The AI will only cut your roster down to 60 if you go into the training camp stage with more than 60. You can have more than 60 during the draft, and you have through late free agency to make your cuts to 60.
While games are being played, you can sign more players and/or be over the salary cap during the midweek stages, but if you don't get that fixed for the game stage then the AI will make cuts for you. (You should get a warning to that effect after the midweek stage). The AI will also sign players if you are less than 45 in a game stage, and if you don't have enough salary cap room to sign players to get up to 45 the AI will make cuts to make room ... that's the worst case that can happen with the AI messing with your roster.
One more point while we're talking about the AI messing with your roster, if you sign one player and cut another player on a game stage (because for some reason you missed the midweek to do that), the newly signed player will be added to your roster as an inactive player, so be sure you have 46 active players not including the one you cut. If you have fewer than 45 active players on a game stage, the AI will rebuild your depth chart.
