Close All Open Apps... Perform a Reset... Try again...
Reset ( No Data will be Lost )
Press and hold the Sleep/Wake button and the Home button at the same time for at least ten seconds, until the Apple logo appears. Release the Buttons.
And there is this...Troubleshooting Messages
I have the same problem with ipad mini . I can send and receive on my iphone 4 no problem . I can connect to the internet on wifi on both no problem . I have turned off and on I have even turned off my iphone imessage to see if it may be blocking the ipad from sending and i still cannot send a message . It starts to send and then hangs at the very end and times out . I have tried everything I found on the internet and this forum . I find if someone sends me an imessage I get it sometimes and I can reply but if I try to start a message it always hangs
I have not done a factory reset yet . That will be a last resort . Is apple doing something on the imessage server that keeps so many people from being able to use it .
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
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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
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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.