On your Mac with Safari open, go to the Safari menu bar click Safari / Empty Cache.
Restart your Mac.
Launch Safari. Try logging into your iCloud account.
If you still have problems, go to Safari / Preferences then select the Privacy tab.
Type apple in the search field top right corner of the window.
Click Remove All > Done.
Quit then relaunch Safari. Try again.
If you still have problems logging in to iCloud quit Safari.
Launch Keychain Access located in /Applications/Utilities
Select Passwords on the left.
Type appleid.apple.com then delete that keychain. You can do that by right or control clicking the keychain then select Delete.
Relaunch Safari. Go to iCloud.com. Login with your user name and password. You should be prompted to save that data to a new keychain. Click Yes when prompted.
I'm having the same problem as described above and am hoping someone has figured this out. I've got an iMac with several user accounts/profiles configured. Two of those user accounts/profiles are set to work with iCloud but each one has a different iCloud login ID. I can login into iCloud from both accounts/profiles when usiing Safari/Firefox, but from one of them I can not login from the System Preferences --> iCloud. The account/profile from which I can login via the System Preferences was the first of the two to be configured to work with iCloud. From the second account/profile, I've attempted to login with that account's iCloud ID as well as the iCloud ID for the other profile. Neither is successful. I get the same error message as those reporting problems in this post: "You can't sign in at this time" and "Unrecognized Apple ID or password".
I am having the same problem. My main account (admin) on my mac does not log into icloud. But when I create a new profile, there is no problem. I do not want to create a new profile otherwise I will need to set up all my preferences and move my address book and calendar across to this account whcih makes no sense. I just need to fix my main account. Any help would be appreciated.