I found some interesting articles instead of the studies, but they should suffice.
The first article by FluentU lists five major advantages in learning multiple languages at the same time: good for your brain, saves time, similarities and differences between languages are clearer, saves you from being bored, and opens tons of opportunities.
Clearly, learning languages is excellent for your brain in so many ways... including structurally improving your brain in three months, as proven in a study within the article by Swedish and German scientists. The study has interpreters put into a three month "military boot camp" for language learning and the results were amazing:
After 90 days of intense training, the scientists, donning their spotless lab coats, came in and again measured their subjects’ brains. They discovered that their interpreters’ hippocampal regions, along with three other areas of the cortex, had grown significantly. The cortical areas increased in their thickness, indicating higher fire power for these areas of the brain.
Also, learning multiple languages can also prevent/slow down Alzheimer's disease and dementia. Learning multiple languages at a time rather than one by one is seriously less time-consuming, especially if you want to be fluent in lots of different languages. It also saves time in life and in studying time:
Imaging that you’re travelling in South America and get lost while on the way to officiate a wedding. You’re standing at an actual crossroad and looking up at two signs. The left one says “Una Via“ (One Way), the one on the right says, “Camino Cerrado Delante“ (Road Closed Ahead). Can you imagine how much driving time you’d save simply knowing what those signs mean?
One way is sequentially and the other is simultaneously. The methods are right in the names. Doing multiple languages at the same time (simultaneously) saves time because, in a way, you’re multitasking. Instead of getting fluent in one language in 1.5 years, you become fluent in 2 languages in 2 years.
You can also take advantage of the differences and similarities between similar languages, such as the romance languages. For example, you can take care of the cognates between languages like French, Italian, and Spanish (arm, fever, and tongue after proper translation):
If you notice, the spelling, meaning and pronunciation of these words (and many, many others) are similar for French, Italian and Spanish, indicating that they have a common etymology.
Cognates are very useful for 2 major things... Vocabulary building... Contextualizing.
You are not getting bored anytime soon. You might be watching Spanish shows one day then Japanese drama shows the next. Being bored will be the least of your problems if it is at all. You probably didn't know this: learning multiple languages can help you get employed, find that perfect dream guy/girl, and understand multiple different cultures:
Over the next decade, Spanish and Chinese speaking skills will be one of the most critical skills sought for by recruiters.
As globalism surges forward, it makes learning multiple languages ever more essential. English isn’t enough anymore. A great number of international corporations aren’t based in English-speaking countries. In addition, US-based corporations are waking up to the fact that in order to flourish in the emerging markets, they have to learn how to say “Ni Hao” and not just “How are you?”
On the romantic end, you’ll be ready to start a bilingual relationship. When your significant other has a different first language than you, what better way is there to express affection than learning romantic phrases in their native tongue? “Sorry” is just different when you say “lo siento.” And who knows, she might just forgive you.
Though not really a credible source, another person's insight might prove valuable, like this discussion on Duolingo (mainly the top reply):
It depends on the person. I will tell you my humble opinion, and tell how it works for me. For me, it is better to deal with multiple languages. For example, I'm learning three languages (Portuguese, Polish, German), and I'm doing well. If I have concentrated on only one language, of course I would progress faster on that language; however, concurrently learning multiple languages makes more improvement on your language knowledge in total, even if it may yield less for each language. Ah, let me explain you with an example: if I learn one language, I will proceed "6 steps" in that language everyday. Not bad. But if I learn two languages, I will move on "4 steps" everyday for each, so "8 steps" daily in total. And even more, if I learn three, I would get further "3 steps" for each, so makes "9 steps" in total. 9 > 8 > 6 . Therefore, the more languages I learn, the more total utility I get. This is my formula, and works well for me. It can be different for other people, of course.
I run an experiment myself. I found out that, I can learn and memorize an average maximum of 20 new words per day if I study just one language. On the other hand, when I learn words from two, three or four different languages, then this number can reach 35, or even 40 words in total, per day.
Also, this is not so relevant to the topic, but just I want to tell because some people have already talked about it: I believe knowing a similar language makes easier to learn the other. For instance, speaking Spanish is a significant advantage for learning Portuguese. Yes, it can be confusing sometimes (Ex: I used to write “é” (the equivalent of Spanish “es” in Portuguese) while I was doing timed practice in Spanish), but after few mistakes, you get to know your mistakes and you don't do it again. And if you know the pattern, you can easily make a Portuguese word by its Spanish equivalent without knowing the Portuguese one, and it'll mostly be correct.
Or, if a person learnt Polish and knows well the seven cases of Polish, then, when that person starts learning German, the four cases of German will be “as easy as pie” for that person, meanwhile many people are struggling and whining about those cases.
Lastly, the more you learn languages, the more your capacity of learning languages increases and the more your brain distinguishes them, consequently you commit less clashes and confusions between languages.
So every person has his/her own efficient way of learning languages. Some people can learn one language in a short time, some can learn many languages at the same time. So just find the best way for you, and learn =)
Finally, this last, not-so-related article quickly discusses some of the disadvantages of this:
Until recently, I never felt the need to learn more than one language at a time. Now that I think about it, my strict “one language only” approach was mainly for three reasons: (1) I was never in a hurry to learn languages; (2) once I chose a language, I was so involved in the learning process that I had no desire to deal with others; and (3) learning only one language at a time gave me the opportunity to polish the ones I previously learned. Now, I’ll discuss these factors in detail.
For starters, rushing through the languages is not helpful. You are going to forget them and you waste all your time learning something you learned so fast it flew past your brain... ouch:
The first six months of the learning process are the most delicate, and one needs to focus on a given language daily and dedicate to it quality work. Time is a luxury that modern men lack, so if you split it between two, three, or even four different activities, the quality of your learning will suffer. Therefore, I personally recommend learning one language at a time.
Next, learning too many at a time will prevent you from having a steady core, which disallows you to have good fluency and quick differentiating of the languages you learned, adding to confusion and question marks:
The language learning process involves the use and memorization of words, structures, and sounds; however, emotions, colors, images, and memories are also involved and contribute to what I call a “language core.” Acquiring this language core is extremely important if one wants to keep the language alive in their head, even long after not having used it. Moreover, it takes time to build a language core, so attempting to speed through the learning process with multiple languages can cause language cores to overlap, or simply prevent even one from forming.
The article also briefly discusses time management and a story to support it:
Picking up a new language doesn’t prevent one from keeping and even refining the ones he previously learned and are obviously at a higher level. The trick is having good time management skills. By trial and error, you want to find a sustainable schedule to learn languages.
In order to better illustrate what I mean by “managing one’s time,” here’s a story I call “The Two Students’ Race.” Two students (both Italian) decide to compete for a challenging and ambitious goal: learning ten foreign languages. The judges give them a ten year “time budget,” and the rest is up to them.
Student A decides that he will pick up two languages every two years. He starts by learning English and Spanish. At the end of the two years, he picks up French and German. While learning French and German, he enjoys conversing with native speakers in both English and Spanish, languages he now speaks with a certain level of fluency. He also loves reading books. By the end of the fourth year, he is relatively fluent in German and French, so he starts learning Portuguese and Swedish. At the same time, he moves to Pariswhere he gets the chance to use the four languages he learned and continue to actively learn Portuguese and Swedish. At the beginning of the sixth year, he decides to learn Mandarin and Romanian. Two years later, he finally picks up Japanese and Dutch. Then ten years later, when time is up, he speaks English, Spanish, French, and German fluently; Portuguese, Swedish, Mandarin and Romanian well; and Japanese and Dutch decently. He has weak languages, but he reached the admirable goal of building a core in more than five languages. Moreover, he will never forget these five languages. He could get rusty in them, but they will quickly come back to him.
Student B goes for the same languages, but opts for a completely different strategy—he starts learning the ten languages all at the same time. Over the years, he lacks the consistency and patience to hold onto all the languages he is learning. Sometimes he learns five, sometimes six or seven, but never ten per day. He might have brought some languages to a good level, but for the most part, his languages are at a basic level. He confuses languages such as Italian and Spanish, and Dutch and German. Many of his languages are destined to regress to a very basic level because the student did not build a linguistic core for any language.
Now, the example is a bit extreme, but it gives you the general idea. People delude themselves into thinking that doing multiple things at the same time will accelerate the learning process when, in fact, it damages the learning process. Remember the story of the tortoise and the hare? Well, be the tortoise. Patience and aiming for a long-term goal will bring you success.
Might rush through languages, lose knowledge
Might not have a good core, disallowing fluency retention and fluency itself
Might not be good for you time-wise