Blue poster, Syndri, suggests three possible resolutions for the login problems:
login server down post in the Technical Support Forums and you may be able to get further help by posting in there.
Please let us know in the comments if this solved your login issues.
Edited to add: Blizzard is not saying it is the fault of the people having problems. They made changes on their end and we just need to update things on our end in order to be able to connect properly again.