That's absolutely true! In ESL teaching (English as a second language teaching) I was reminded many times that "good writing is good reading", "good writing is good speaking", etc.
I would describe it this way:
Reading(as a passive skill) exposes you to a big amount of vocab and grammar, which helps you identify vocabulary and grammar patterns in speech, it also makes writing easier since you now have a big arrange of different words you can choose from.
Of course this doesn't mean that if you only practice one part of language you'll master the other three parts. It really means that by practicing, for example reading, you can get a good start for the other areas.
An analogy that I think might be useful, is thinking about how one sport can help you with another one. For example, a runner could have a good start with swimming, he will NOT became an expert swimmer because he is a runner. It might be easier, but he still needs to practice to become a better swimmer.
It's the same with language learning. If you focus on one area this will give you a good boost when you start with another area, things might start to click more fast in your mind for example; but if you want to be a good language learner you need to use the four skills: listening, speaking, reading, and writing.
Thank you for sharing your question, this has gave me an idea for a blog post. :D