There isn't much information about Bogie playing chess. I've read somewhere that he used to play for money in the early thirties when he was just another starving actor.

The chess scene in the beginning of Casablanca was actually Bogart's idea so it seem very likely that he was a chessplayer. If the game below is authentic he also invented his own opening, it seems. Although, nowadays it would probably take some HB (Swedish moonshine) to get someone to play the HB Gambit!