Using FaceTime http://support.apple.com/kb/ht4319
Troubleshooting FaceTime http://support.apple.com/kb/TS3367
The Complete Guide to FaceTime + iMessage: Setup, Use, and Troubleshooting
Troubleshooting FaceTime and iMessage activation
iOS: FaceTime is 'Unable to verify email because it is in use'
Using FaceTime and iMessage behind a firewall
iOS: About Messages
Set up iMessage
iOS 6 and OS X Mountain Lion: Link your phone number and Apple ID for use with FaceTime and iMessage
How to Set Up & Use iMessage on iPhone, iPad, & iPod touch with iOS
Troubleshooting iMessage Issues: Some Useful Tips You Should Try
Setting Up Multiple iOS Devices for iMessage and Facetime
FaceTime and iMessage not accepting Apple ID password
Unable to use FaceTime and iMessage with my apple ID
FaceTime, Game Center, Messages: Troubleshooting sign in issues
How to Block Someone on FaceTime
My Facetime Doesn't Ring
To send messages to non-Apple devices, check out the TextFree app https://itunes.apple.com/us/app/text-free-textfree-sms-real/id399355755?mt=8
How to Send SMS from iPad
You can check the status of the FaceTime/iMessage servers at this link.
I am having the exact same problem. I've been searching forum after forum for 3 days now and tried several so called fixes without any luck. I've also read through the applicable links provided by Texas Mac Man and had no luck. The only other area this issue seems to be causing a problem is in Find My IPhone which will display my MacBook Air and iPhone but not my iPad. The issue started as soon as I updated to 6.1.3.
Change DNS to 126.96.36.199
Reset all settings
Make sure date and time set automatically on
Make sure region set properly
All other functionality works fine including iTunes and the App Store.
Immediately after writing these remarks I gave it one last try and I was able to sign into IMessage and Facetime on my Ipad and Iphone 5. What did I do? Absolutely nothing. I reset nothing, changed nothing, all of my contact location information was in sync from before, the time setting was on automatic. The only thing I did was to try repeatedly and lose two hours reading forums on the internet. Oh, yes. I did do one thing that seemed to have worked. I vented my frustration on this forum. Maybe the ghost of Steve Jobs heard me. LOL