This may be really stupid and simple and make you think "Really!? That's it!?" but I was literally just having the same problem (and the servers weren't down) and I fixed it
I clicked the "Play Game" tab and erased my password + username, then rewrote them.
I opened the game back up, erased my username and unchecked the box that allowed you to remember the username, then click it again and retyped my username + password and logged in, and it worked.
Have you tried something like this?
And another one, have you tried switching browsers?
Example: A weak one (like Internet Explorer) to a strong one (like Firefox or Chrome).