Well, Spock definitely wasn't a cadet. Neither was Chekov, in spite of his age. And Scotty wasn't, if you want to include him. I'd imagine there were a few other officers on board as well. Most of the crew were probably cadets, true, but not all of them -- frankly, I'd be very surprised if Starfleet had all these ships in working order and nothing but cadets (albeit soon-to-be-graduated cadets) to staff them with. It seems rather implausible.