iCloud could not connect only with my Apple ID

When I try to connect to my iCloud account with, for example with Messages, it says that it couldn't connect, that I need to verify my network connection and try again,

My network connection is working fine, my Apple Id is also working fine (I have tested it successfully on icloud.com and it is also working fine on my iPhone).

When I go to the system settings, select iCloud, my account is still connected (because this issue is new, I was connected a long time ago), but when I check my account details, it asks my password and the attempted connection never ends. No error message, nothing except an infinite loading...

Do you have any clue on that issue?

It was working fine on Yosemite, then suddenly Messages stopped working, I disconnected my account and then tried to connect again and it never happened. I have updated to El Capitan, but the problem still the same.

It seems the problem is only affecting my Apple ID, because it is not happening with other Apple ID that I have tried.

Do I need to contact Apple for that since it is only happening with my Apple ID?

Thanks for any advice!