I can't sign into iMessage or FaceTime

open the "Keychain Access" app from the Applications > Utilities folder (or from spotlight search).

once the app opens, click on the "Keychain Access" menu in the top left corner, and then select "preferences".

In the window that opens, click the "Reset My Default Keychain" button.

CAUTION: This will delete all your saved forms and passwords on your computer. Make sure you know them before proceeding because you will be prompted to re-login to every single website and application where you had previously saved info.

Restart your computer. Viola. You are fixed. 🙂

Here is a solution that worked for me, after nothing else would. Log out and back in to iTunes on your Mac/PC.

I had exactly the same problems as above, and no amount of network changes or even a full restore would help.

Finally, I went back to my PC, launched iTunes, logged OUT of that, then signed in again, this time with the new password.

Then I had to go back, and properly close Facetime and iMessages and sign in. It worked fine after that.

(To properly close an application, double click the home button, then hold down on the facetime app logo in the multitasking bar at the bottom, then press the tiny ➖ symbol.)

Best of luck.