A question that’s never asked is whether you can improve too much, probably because most people have trouble improving at all. Actually it’s quite an interesting and relevant one as players who get over 2200, especially titled players, have difficulty finding suitable opposition. This is quite natural because players with this sort of rating tend to be quite rare. But it can be disappointing to spend a lifetime honing your skills, only to discover that you have trouble using them.
Travelling to international tournaments is the solution of course, but life can get in the way of such things. Internet chess is imperfect at best, though many end up taking this route. As for playing ‘down’ in local events, it’s not really satisfactory at all. You develop bad habits by playing weaker opposition, the playing conditions tend to be poor and people can even become resentful and hostile if you dare to play for a rival team!
I think this is why many Grandmasters have simple stopped playing; they loved the time they were globetrotting professionals too much to settle for being amateurs again. Actually this is how I feel myself, though I may be playing in some local events with my son as he develops his game.
So what’s the optimal level you should aim for for chess to remain an enjoyable hobby? Well probably around 2200, which is not too high for club chess whilst allowing you the possibility of international tournaments and playing against titled professionals. Probably it’s still OK to be a bit higher than that, maybe even somewhere in the 2300s. But after that you’re no longer in the amateur ranks even if chess is still a hobby for you.
How can you stop improving and just maintain this sort of level? Well probably it’s best not to study too much and take some time off the game if you feel chess strength surging within you. In the worst case you can adopt some dubious openings, for example the Latvian Gambit against 1.e4 and the Grob as White. Though this might take the fun out of the game if you’ve reached a level in which the pieces and pawns become one with your psyche and dubious play becomes an offence to the soul!