Mail Server Settings for Email Clients

There must be something wrong with my iCloud account but I can't figure out what it could be:

I can't add my iCloud account to third party clients (Postbox, Airmail & MailPilot) I get error messages.

I succesfully added my cousin's iCloud account in those apps, on my own iMac in all 3 apps, no problem at all, yet It doesn't work with my account even though I'm using the exact same settings in both cases!

I'm using the following settings

IMAP Server: imap.mail.me.com

IMAP Port: 993

IMAP Connection Type: TLS / SSL

IMAP Authentication: Auto Detect

SMTP Server: smtp.mail.me.com

SMTP Port: 587

SMTP Connection Type: Start TLS

SMTP Authentication: Auto Detect

I did peruse this page : http://support.apple.com/kb/ht4864 and tried many changes, no can do! I also tried to replace ' imap.mail.me.com ' and ' smtp.mail.me.com ' with ' p07-imap.mail.me.com ' and ' p07-smtp.mail.me.com ' (I found ' p07 ' in Mail.app settings). I also tried p04, p99... Nothing seems to work.

But then again those 3 mail clients work instantly with anyone else's iClound account, on my very iMac! (which sports OS X 10.9.2)

Other than that, no problem using my iCloud account on MAS, iTunes Store, iTunes, iOS. iCloud.com...