OTB Question

Is it possible to not realize that under check when you're playing over board and make an illegal move?

it may be possible or maybe not

Imagine you're playing an opponent over the board at a park.. how do you know your opponent would notice?

Of course it's possible. This sort of thing can happen with lower rated players like myself. I would know that I'm in check playing a standard game of course but I could certainly miss a check playing a 3/0 game and forget about bullet.

Let's say you're playing a hustler at the park and he tells you "you just made an illegal move" avoiding a check even when you didn't? But you don't even remember it and they take your money

In an USCF game, unnoticed illegal moves are binding (e.g. game ends and you lose, later you noticed an illegal move in the game doesn't change the game result).

If you claim illegal move during a game (I'm not sure, but it might have to be within some move limit), then you can rewind the game to prior to the illegal move.  If nobody is keeping score, then illegal moves cannot be claimed.

Yes, it's possible if both players do not realize it's illegal and I've had to rule as a Tournament Director - usually the illegal move was castling through a check in high school tournaments.

According US Chess Federation rule 11A, if it is discovered that the illegal move was made within either player's last 10 moves, the last legal position will be reinstated but no time adjustment is allowed according to rules 11H and 16R. But since the Tournament Director is allowed a certain amount of discretion, if there was less than 3 minutes left on either clock, I used to add an equal amount of time to each clock so there was 3 minutes on the short-time-left clock - but I only ever overrode the official rules if all the coaches of the teams present agreed.  Otherwise, a player might be in potential time trouble due to the illegal move not being noticed.

Once the last legal position is reinstated, the piece first picked up to make the illegal move MUST be moved if a legal move is possible according to the Touch Move Rule.

According to rule 11B, if more than 10 moves have been made since the illegal move, the game will continue without returning to the last legal position.

If the players' scorepads are messed up or disagree to the point where the last legal position can not be certainly determined, the game will continue without returning to the last legal position.