I had this problem, and none of the previous answers worked for me. After testing several different approaches, I finally discovered what the problem was:
* Your password must have at least 1 uppercase letter and 1 numeral in it.
*
The reason that new accounts work is because the process Apple set up for creating new accounts requires that you have at least 1 uppercase letter in your password. Older accounts, on the other hand, don't require having an uppercase letter.
Try these steps:
1. Go to:
http://appleid.apple.com/
2. Click on *Manage your account* and login.
3. Click on *Password and Security*.
4. Click on *Change Password*.
5. Change your password. *Make sure your password has at least 1 uppercase letter and 1 numeral in it!*
6. Click on the *Save Changes* button.
7. Finally, go back to your iPhone or iPod Touch and try your new password in FaceTime, and voilà!
Please let me know if this works for any of you. I am curious to know if this helps people out.
Good luck!