Similar to what Matthew said-
I don't think that I learned the nuts and bolts of how to stage manage in a class in college. I learned how to make a run sheet, how to take good blocking notes, when to speak up in rehearsal vs when to keep my mouth shut, how to deal with an actor having a melt down, all of that by actually being a stage manager. That's not to say that my stage management class didn't help with those things, but I would say that 80-90% of it was learned by SMing shows in college, and interning at various theaters. I'm still learning things with every show I work on now.
HOWEVER.
What I DID learn in college was all of the stuff that seemed at the time to be a waste of time. Why do I have to take Advanced Acting? Directing- do I really need that class? UGH SCRIPT ANALYSIS WHYYYYYYY?! One theater history class wasn't enough??
It was those classes that I didn't enjoy at the time that taught me things I would have found more difficult to learn "on the job." The most invaluable, for me at least, was all of those darn acting classes- I HATE acting, but having to take it and do it and read about it gave me not only an appreciation and respect for actors, but also makes me better able to communicate with them, to give them notes, to understand what the director means when s/he is giving notes, to be able to tell when an actor's preferred process isn't going to mesh well with the way the director is working. The director wants things "more Brechtian," that actor studied Meisner technique, blah blah blah. Yes, I get better at that by doing it, but not having that baseline vocabulary to start out with would be killer.
Same with theater history- yeah, you could SM a commedia piece without knowing anything about it, but how much better would you be if you know the difference between the Dottore & Arlecchino starting out? If you already know what to expect in terms of physicality, blocking, combat? It'll be a lot easier to SM a Shakespeare piece if you've studied how to analyse the language already.