For example

BYU has a Nike deal

BYU Swimming has a Mizuno sponsorship. The team wears both because Nike can't offer anything to the program outside of basic apparel and training gear

If a BYU swimmer got an NIL deal with a high reputable swim brand, let's say Speedo, all of their BYU swimming training gear would be Speedo, but their BYU non training gear/apparel would be nike