I've been thinking about doing just that. I've been looking at agglutinating languages like Korean and Hungarian. Problem is, I've never managed to truly learn a language unless I had a use for it. I've only ever learned German well, because I used to listen to NDH metal all the time.
I would really like to get a deeper understanding of how these languages work, but I don't see myself being able to actually learn a language that I have absolutely no use for. I can't even complete a German course on Duolingo because that site's lessons are the only thing I ever use German for anymore! I never speak it or write it, and I rarely ever hear or see it anymore.
Learning languages can increase your linguistic knowledge. German, for instance, helped me to understand how a case system is used. I don't know if I could ever have understood a Latin-like case system if I didn't know German.
Do professional linguists do stuff like this? Note that I don't have a degree, I'm entirely self-taught. I've been studying all I can on linguistics for over a decade now, but I've mostly focused on phonology (I got into linguistics so I could make an auxlang, I've since given up on that goal). Recently I've been trying to focus more on grammar, but like I said, I don't see myself being able to truly understand some things when I can't see examples of how its used.