Apple ID verification failed

Signing onto iPhone I am asked for Apple ID/psswd. I enter and get verification failed. I change psswd and go back and enter new psswd as confirmed by email from Apple.

Verification Failed

The application you have selected does not exist.

Well, I was trying to log into iCloud, an application which does exist.

My ID/psswd worked on logging on from my Mac and into this forum so it's the correct pairing.

How do I get my iPhone to accept? And, yes, I have turned off wifi and tried over dataplan to the same outcome.

Thank you.