This fix is very, very similar to the fix for iMessage. Both problems can be fixed by the same solution.
If you are trying to sign into FaceTime using your Apple ID and it says "Could not sign in. Please check your network connection and try again," try these simple tips to fix the problem.
Step 1) On your iDevice, go to the Settings App.
Step 2) Select Wi-Fi.
Step 3) Select the network that you are currently connected to and click on the blue and white arrow on the right of that network.
Step 4) Click on DNS and change the current value to 18.104.22.168,22.214.171.124
Step 5) Open Safari and go to any website just to make sure the settings are working.
Step 6) Open up FaceTime and you should now be able to register!
Also, make sure that iMessage is not disabled in the restrictions section under Settings. iMessage and FaceTime are linked and disabling one will in fact disable the other.
Another attempt at a solution, under Settings > General > Date & Time, make sure that Set Automatically is turned to "ON."
If you are still experiencing problems, you will have to reset your iDevice and setup as a new device. Cumbersome, but it works.