No. For production it comes down to talent and spending enough time to learn your particular production tool (the DAW) and other related technology inside and out.
There are so many unemployed people that graduated from music production/engineering schools. I don't think it's worth the money. Intern at a studio and try to work your way up.
I would say yes to mixing and mastering, though. You really have to know the technical aspects inside and out. But again, with enough studio time and the right mentor, you could learn this without having to pay for college.
Time and talent. How much time are you willing to put in and how much talent do you have? You can't teach a producer who has the right ear for melody or has the right feeling in their mind to make your arrangement sound good. Rick Rubin, Dr. Dre, Kanye West, Skrillex, Mutt Lange, Bob Rock, Butch Vig... the list goes ON and ON about folks who never went to school for production.
Too many people try to "break down" and make this stuff scientific. It is if you're messing with mastering and mixing to an extent, but talent is not a mathematical formula.
I think finding the right mentor is more important than going to music production/engineering school.