I've been having problems with signing in all day and was just able to fix it, but I'm not really sure why.
I'm on an iMac and I was connected to the internet through an ethernet cable to my router/switch. I turned off the Ethernet (from Settings -> Network) and turned on Airport WiFi then connected to my neighbors unsecured WiFi.
Immediately I was able to sign in no problem and use FaceTime no problem, tested it with my iPhone 4. I figured somewhere with my router/switch was the problem.
However, I then turned Airport WiFi back off and turned the Ethernet back on and FaceTime still works. I've signed out, signed back in, quit the program, restarted...everything is still working. I don't know why it fixed it but it did it seems.