In the movie, I don't know what level they were, but I would suspect AA since their bus was decent, and it is a good crossroads for players on their way up and down. Jumping from AA to the bigs isn't unheard of... Pitchers with good "stuff" still have it regardless of the quality of the opposition. So Robbins's character could have gone to AAA but instead went directly to the show.

In real life at the time, the Durham Bulls were a single-A affiliate of the Atlanta Braves. They are now a AAA affiliate of Tampa Bay.