How do I resolve 'This Mac can't connect to iCloud' error with incorrect password message?

Hello,

I am getting an error message on this MacBook saying 'This Mac can't connect to iCloud because of a problem with myemailaddress@me.com . It says 'Open Account settings to fix this problem.' In Account Settings it asks for my email address and password. I enter the password and it says, 'Your identity couldn't be verified. Check your network connection and try again.' The network connection hasn't changed, and I am logged in to iCloud on this computer via my browser. I re-entered the password, and it then said, 'Your password has been entered incorrectly more than once. For your protection you will need to retrieve it before continuing.' It then gives me an option to reset my password. I reset the password, confirm the password. And then a few minutes later I get the message 'This Mac can't connect to iCloud because of a problem with myemailaddress@me.com' again, and then the whole process starts again.'

I have changed my password twice now. Still the same problem. What do I do now?

[Re-Titled by Moderator]

Original Title: iCloud login problems