I have an iPod touch 6th generation but I can't sign in to Face Time or iMessage
I've had similar experiences with my iPod 6th Gen too. I suggest that you check your network, or WiFi and try again. If it is a network issue, then you might have to wait until it is resolved. I fixed mine, after figuring out that it was a faulty WiFi and not the iPod (the Wifi option was on) or the service. Hope this helps!