All of a sudden today my iPad stopped syncing iMessages. I turned iMessage and Facetime on and off, and reset my device, several times but nothing worked. So I signed out of iCloud, reset my device, then tried to login again to iCloud. The Settings screen has been stuck at "Verifying" for 15 minutes, and I have repeated this process a few times with the same ending. While it's "Verifying" the Settings screen is frozen. I can't scroll or tap anywhere, but screen rotation still works and I can go to the App gallery and all apps work fine. FYI my internet is working just fine as I am writing this post, and I can also login through icloud.com just fine. Note I have 2-step verification on. Thank you.
Asking another person to call you, if the call is successful the bug goes away. At least that happen in my case, after upgrading to 10.3.1 suddenly my wife called and voilá! eveything back to normal.
I think this is a bug. I solved it by "baiting" the system by inputting my password through App Store. That did the trick. Unfortunately I dont know how to reproduce this bug.