I dont know personally when other people move to new instruments but i can tell you mine.
I took a minor in Jazz Piano in College (Major-ed in Chemical Eng) and thus music for me is just a hobby and a passion. After college and playing for a few months, I got kinda bored playing the piano and i went back to saxophone (used to play it in high school, never touched since for 6 years). I was an alto man, and i went back to play tenor and soprano instead.
Right now, I am moving on to another instrument, Bass Guitar, not because i am bored with the saxophone but rather, the reason why i am diversifying is to improve my jazz musicality and hopefully improve my piano playing.
Everyone has reasons to move on to another reason, and what i mentioned above is the reason. Some do it because they need to pay the bills, others do it because the current instrument has certain problems (cannot play it too well or get bored).
What i advice you is to really work on 1 instrument first (like the saxophone) and make it your primary instrument. Really know it very well and when you move to another instrument, you can take the knowledge of that instrument into the new one. Personally, every new instrument i pick up improves my improv and jazz musicality on my main instrument, piano.
So figure what you want to do.