When something is intolerably hot, slow, noisy, painful, or unfair, it has gone past the point where you can still live with it. You are not only unhappy — you cannot accept it, and you may not be able to deal with it either. It is a strong adverb, used of heat, pain, waiting, rules, and other people's behaviour. If a class is intolerably boring, you almost mean you could not sit through it. Writers like it because it is sharper than a plain “very”: it says “too much to bear.” You will hear it more in careful speech and writing than in quick chat, where someone might just say “I can’t stand it.” Though actually, it only fits a bad kind of “too much.” “Intolerably kind” sounds odd, because kindness is not something you cannot accept.