Just replying because I had this problem the last few days. I tried to contact CS but they weren't able to help... so I decided to investigate it on my own.
I could log into the game no problem and I could log into the sites on my mobile phone... but not on my PC, so I figured it must be a problem specific to my PC (client-side) rather than Anet (server-side).
It turns out that my antivirus, Bitdefender, was blocking certain websites that were hosing javascripts used for the login forms, making the form fail on submission. I tried to add them as exclusions but I think there's too many URLs to add (or the free edition of Bitdefender doesn't handle exclusions very well) so I ended up just uninstalling Bitdefender altogether and I was instantly able to log in again. It's possible Bitdefender had a recent update or something, idk. Either way, I'm using something else now. :\