So, when do you call it quits? Unless the type itself is an art piece, the fewer the better! A good design can happen with one font. One. By changing the attributes (bold, italics, condensed, etc.) but keeping the same font, a design looks clean and professional.
Two is fine, assuming that the fonts are assigned in a way that makes sense. A different font for the headline, or a call-out can look really nice. There are certain characteristics of different fonts that do work well together and can compliment each other, just as those characteristics can also not work well together. If the fonts are too similar, it looks weird.
Usually, combining a serif and sans serif font is a good place to start. A serif is the little pokey-uppy or hangy-downy thing that some fonts have. A sans serif font does not have those.
Four? That's just crazy talk!
I recently saw a poster advertising an upcoming event for a local charity. The poster had no less than 6 different fonts on it, and it's a jumbled mess. It's difficult to read and there is no flow or sense of balance to it at all. And sadly, it looks amateurish.
Even famed artist Leonardo da Vinci was quoted as saying “Simplicity is the ultimate sophistication,” although I'm sure he said it in Italian. But I'm sure he meant Keep it Simple, Stupid!