Blizzard representatives are on the case, and while they haven't fixed the exact problem, they have found a workaround: This is only happening with the World of Warcraft specific login servers, and thus it's only happening to World of Warcraft specific logins. If you merge your World of Warcraft account into a Battle.net account, you will be able to log in.
It's no fix, but if you're desperate enough to play and haven't yet merged your account over, now is a pretty good time to do it. Personally, I'm not sure why you wouldn't do it anyway, but that's just my preference talking. If this login issue gets worse before it gets better, we'll be sure to let you know.
Update: This issue appears to be cleared up, but please let us (and Blizzard of course) know if that's not the case!