iCloud dot com doesn't recognize my login

This is weird. I went to icloud.com and it wouldn't accept my Apple login despite multiple attempts, slowly and deliberately entering the proper password. I even clicked on forgot password and changed my Apple account password to see if I could goose it into cooperating but that didn't work either.

Of course I had to reenter my new password into all my other Apple stuff like iTunes and mobile devices, etc. The new password works now on all of those things (then again, the old password had continued to work before I changed it too). But for some reason, icloud.com won't let me into the website to deal with my icloud apps.

What gives?

alan b