It took me two and a half years of serious several-days-a-week studying with coaches and tournament pkay to break get past the 1500, as a teenager!
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1600 is a great rating! With that, you can probably beat anybody at work, anybody in your family, anybody at most any party you go to.
Do you realize, in the US, there are entire rural counties, do not have anybody rated that high?
Buck up! Only question you need to ask yourself is, are you enjoying yourself playing chess.
If the answer is yes, cograts! You have a hobby you can easily do until your 80. Sure will beat bingo or shuffleboard!
I have to play a lot of chess for job reasons and thus had to get into it at my old age of 40. I read a lot of books, learned the tactics, basic positional play, endgame and openings. I meet players rated 2000+ regularly. They discuss with me how to learn chess, analyse my games, show me games and openings, etc.
So I have been trying this for more than a year now, as much as time allows and I can bring the energy into.
But yet, I do not progress that much. E.g. I do tactics puzzles a lot, but yet cannot get over 1400 at the tactics trainer here. It is very hard for me to remember opening theory, and miss tactical opportunities.
I have a 1600 something rating at the "online" games here (not that this means a lot, I guess) but I do not play like this anymore since I realized it does not benefit my visualisation skills.
Am I simply to old to get into chess at age 40?