Had the same issue yesterday. Changed my iCloud password and then it asked me the verification code from two-step authentication. I did get codes but it was always saying incorrect. Since I tried many times, Apple locked my account. Contacted them and the senior advisor told me to wait 24hours and try again. I asked him the reason, he said my internet connection could be slow. I don't think that is the case. I am sure there is an issue with Apple servers.
I still have few more hours, will try this afternoon.
03-31-2017 02:51 AM