I recently got my iPod Touch, and a couple days ago, I was signed out of FaceTime and iMessage (not by my own doing) and I haven't been able to log back in. I have already tried changing my DNS to 126.96.36.199 and 188.8.131.52, 184.108.40.206, I have logged in and out of my Apple ID on my iPod Touch, I have changed my email address and then changed it back, and I have used a different WiFi connection. The message that pops up (after it fails to log in) says, 'FaceTime Activation Could not sign in. Please check your network connection and try again.' This happens for iMessage, as well. I am using the right password and email, and I do not use an iCloud account. What should I do to be able to log in again and keep my email the same?
I will try restoring the iPod itself. It sounds like a software issue, maybe about cache.
There's also another possibilities about your network routers. It might be blocking those ports or whatever. Try with an Apple one.
reset your setting then make sure your Bluetooth is not on. This worked for me. Or just try the Bluetooth first!
By chance do you have a cover/case on you iPod? Go into the room where your wi-fi router is and try again. If you do have a cover on it, remove it first to activate it and then if it works you should be able to place the cover back on still use it. I have a Survivor case on mine and that was blocking the signal needed to activate. It didn't get in the way of itunes, internet connection, or anything else. But for some reason it did for FaceTime and iMessages. Good luck!