Created an account for this reason. Having the same issue here. 10.9.5 on Mid 2012 Mac Book Pro. Everything was working fine and suddenly unable to connect to the server. Tried logging out of icloud via settings and back in and of course that removed all existing downloaded e-mails from the mail app. Since everyone experienced this at same time likely apple's fault. Wish they could notify people prior to doing these changes.
"The server returned the error: The connection to the server “p36-imap.mail.me.com” on port 993 timed out."
"The server returned the error: The connection to the server “p36-imap.mail.me.com” on port 993 timed out."