While your elementary school English teacher probably told you to never use them, double negatives actually do have a place in our discourse. (Those teachers should have known to never say never, not ever, not ever.) Since grammar girl has a relatively lack-luster article on the topic, here
Example: "I thought she would have liked me, or at least not disliked me, but it seems I was mistaken."
Example: "But of course I'm excited! I can't not be excited!"
Example: "I don't like that one. I don't like that one. I don't...hmm. Well, I don't dislike that one."
Example: "Unlike previous concerns, this is no insignificant matter."
Example: "I don't dislike your course, professor; I just have other courses which interest me more."
Example: "I am not impressed, nor am I unimpressed."
What we're talking about is using the double negative for well-thought-out strategic rhetorical purposes. When people say not to use double negatives, they're trying to avoid:
Example: "Isn't it true that you were never there that night?" (Literally: Is it true that you were there?)
2) Failure to convey proper meaning
Example: "I don't want nothing to eat...leave me alone!" (Literally: I want something to eat.)
3) Useless words
Example "A not unblack dog was chasing a not unsmall rabbit across a not ungreen field." - Orwell
Well, there you have it. So the next time you can't avoid slipping a double negative into something you've written, take a moment to evaluate what you're doing. If what you're saying can be said more simply with a positive, without the loss of meaning, go for it. Otherwise, know that, just as two wrongs make something very different from a right, it's not impossible to produce with negatives something more than a mere positive could ever provide.