Yes, there have been studies showing some disadvantages.
Bialystok (2008) notes that
It is now well documented that bilinguals generally control a smaller vocabulary in each language than monolinguals (Oller and Eilers, 2002; Perani et al., 2003; Portocarrero, Burright and Donovick, 2007).
This has been found to be true in children, adolescents, and adults, although the author notes that
The bilingual deficits in lexical access and retrieval persist with aging (Gollan, Fennema-Notestine, Montoya and Jernigan, 2007), although a study by Gollan, Montoya, Cera, and Sandoval (2008) showed that the effects of aging
interacted with word frequency in that older bilinguals demonstrated a smaller deficit for low-frequencywords.
Protocarrero et al. (2007) come to the same conclusion as Bialystok and the other studies cited.
While the reasons for this are not known, there are two competing theories:
- Bilinguals, on average, use each language less often, and therefore reinforcement of different links is weaker in each language. Mindt et al. (2008) touch on this hypothesis a little.
- Bilinguals may have worse abilities to recall words because their L2 language may have been learned significantly later than their L1.
Note that older studies believed that bilingualism caused a reduction in IQ (see e.g. Darcy (1953)). However, these studies were flawed and are no longer accepted as accurate.
There seems to be a consensus that bilingualism is, on the whole, an advantage, and various benefits are constantly touted. I'm not disputing those findings, but it's interesting to pay attention to the fact that there may be a bias towards studies that show that bilingualism is a positive thing, as this New Yorker article says (the cited study is de Bruin et al. (2014)).
A relevant phenomenon is code-switching, when a multilingual switches between multiple languages in the same sentence or conversation. The Linguistic Society of America says that this is not an issue for children talking bilingually at home. However, it makes sense that it could be disorienting in a conversation with people who speak only one language. That said, this is not a likely common problem, nor a severe one.