Not at AIM. The sport is predesignated, and you play with your company at whatever the sport is that day. Think college intermurals, for the pride of the company. It gets pretty rowdy with the Cadre trying to coach their AIMsters into beating the other companies, and by extension, the Cade's classmates.
However, I do remember being able to meet the coaches of all the sports during AIM, if that helps.