Yes, to imply low intelligence. I think it's an American term? Though I'm not entirely sure where I picked it up from.

Seems a bit odd to use it as an insult though, as quite a few people (myself included) are more or less 'mouth breathers' because of medical conditions.

We all breath through our mouths. I could have described him as having 'slopey shoulders', I doubt the progenitor of that insult intended people with back problems to take insult, though I guess some may take it too literally.