Is there a word to describe one who brags by complaining?

An applicable neologism is humblebrag, a boast couched in self-deprecation. Henry Alford describes it as the false modesty of a plea intended to be met with both awe and sympathy. Humblebrag is attributed to Harris Wittels, a writer for the NBC sitcom Parks and Recreation who Tweets celebrity examples at that handle and has collected them into a book.

A less popular term from a different U.S. sitcom is backdoor brag, from a 2008 episode of 30 Rock. While not as buzzworthy a term, the example used in the episode matches one of yours:

It’s sneaking something wonderful about yourself into everyday conversation… Like when I tell people it’s hard for me to watch American Idol, because I have perfect pitch.”

Update: J.P. Freire in the American Spectator (a politically conservative U.S. magazine) proposes whinebrag, photobrag, busybrag, sorrybrag, and emobrag to capture additional flavors of show-offy self-deprecation. Who knows if any of them will gain currency; he notes the overlap of whinebrag with first-world problems a.k.a. white whines, and the Harvard Business Review lamented the busybrag months ago.

I think the more traditional terms might be false modesty or false humility, defined as:

Noun: behaviour in which a person pretends to have a low opinion of their own abilities or achievements

Here is an interesting article from The New York Times, which actually uses @choster's term humblebrag in the title.