A little bit of googling makes it seem like auth tokens are not sent it plaintext over HTTPS but are authenticated using challenge response – http://forums.dropbox.com/topic.php?id=47952 The WWW site may differ.
Obviously that would work, but not if you're using Challenge-response authentication. In general, I don't think people bother with now that when using https.
Assuming they use straight up salted sha256, my five year old core2 laptop does at least 10,000 per second, per core. They could check every user for the top 10k passwords for a few hundred bucks of EC2 time.