Problem login to icloud with same credentials that work on iPhone, MBP, iPad

I tried to find similar problem, but I couldn't.

I have madamov@mac.com AppleID that I use for iCloud (the one I am posting from is completelty separate because it is connected to my Trainer resources) on my MacBook Pro, mini and iPad for some time. Few days ago I bought iPhone 5 and I wanted to login to iCloud from it, I couldn't so I reset my password. After some problems, I managed to login from iPhone, iPad na MacBook Pro, and I could login to www.icloud.com using AppleID madamov@mac.com with new password.

Yesterday I tried to login from my Mac mini and I couldn't. Then I tried login to www.icloud.com from several browser and from all devices, I even tried from my firend's Windows machine. However, my MacBook Pro, iPhone and iPad uses iCloud without any problems, the password is also working since I could get to a dialog displayed on included screenshot, and in order to do that you have to choose to manage iCloud storage from iCloud preferences and then to click on "View account" button which asks for password.

I tried login in with madamov@mac.com, madamov@me.com, madamov@icloud.com and just plain madamov as AppleID, no luck. Anyone had a similar problem?