I'm though round-robin chess qualifications play for a set number of games. Most organizations simply don't have the money or space to organize a tournament with potentially unlimited gametime, and tie-breakers only happen in absolutely top-level play.
Then again I don't follow modern chess that much, so I'm not really sure about that.
Until 1986, world championship matches were played for an unlimited number of games or until one player had 6 wins. From then on, until somewhere in the 90's they decided to play a fixed number of games, and ended the match when one of the players had the required number of points to win (i.e. if one of the players had 5.5/7 out of 10 games, the remaining games wouldn't be played). These matches were still required to have a decisive result, so in case of draw, the champion won. In recent times, they play a fixed number of games, and in case of a draw, they'll go on to faster and faster time controls and ultimately armageddon.
Tie breakers can happen at all levels.
But this is a different issue altogether, what was being discussed was the result of a game, not of the tournament.