Absolutely, we all reach a peak. There is also the "law of diminishing returns". It is easier to improve when just learning the game. Read a book for a few hours then play other beginners and your improvement is immediate and immense. After advancing for several years, a few hours of study, practice and analysis will result in negligible improvement.
And then, of course, as Denver pointed out earlier, there is the inevitable decline. Enjoy the ride in either direction for as long as you can. 
If you keep at chess, then why would you stop improving? 
Obviously, some physical factors bring down chess players in older age (declining health, declining memory, tactical vision dropping etc.), but some players seem to keep improving - regardless of age. Take Mikhail Tal for instance. He beat his first GM at age 18, became a chess grandmaster himself by age 21 and won the World Chess Championship at age 24 (1960), yet didn't reach his peak chess rating (2700+) until age 44!
Unfortunately Tal passed away at age 55 due to declining health and eventually kidney failure; his renowned chain smoking and drinking undoubtedly didn't help. Who knows just how high his rating may have become if he was still alive today!
after you drop dead your chess is done so then for a fact