When can a redundant "respectively" be dropped?

I don't think it's clearly obvious in all situations that the pairs match up. I guess the main purpose of respectively is precisely to remove any potential ambiguity. Observe:

• Jane and Joe are beautiful and smart.
• Jane and Joe are beautiful and smart, respectively.
  
  
• Japanese, Italians, and Russians are good at making sake, grappa, and vodka.
• Japanese, Italians, and Russians are good at making sake, grappa, and vodka, respectively.

So I suppose you should always double-check whether you would still be saying the same thing if you dropped respectively (or any word, for that matter).

The "respectively" in those sentences isn't redundant. You might argue that it's unnecessary (in which case I'd disagree^*), but "redundant" means that the word is repeating something you've already said. Think of the classic examples of redundancy "ATM machine" or "PIN number".

^* As RegDwight has demonstrated, when you're listing nouns and their adjectives separately, the "respectively" is actually necessary. Otherwise, the reader has no reason to think there's a one-to-one correspondence between the lists.