Q: I have an icloud error "The server returned the error: The connection to host p08-imap.mail.me.com on port 993 failed."

Did you try going to System Preferences>iCloud, unchecking Mail, then rechecking Mail?  (This removes, then recreates the email account.)

I believe that's an indication that there is an error receiving mail, but if you have any drafts or email in your outgoing mailbox, try deleting them.  Apple's troubleshooting steps for this are (from http://support.apple.com/kb/TS4002):

Cannot receive mail in OS X Mail

If you use OS X Mail, look at the name of your iCloud account on the left side of the main Mail window. If your iCloud account name is dim and has a lightning bolt next to it, your account is offline. To resolve this, make sure your computer is connected to the Internet. Then choose Go Online from the Mailbox menu.

If taking your iCloud account online doesn't resolve the issue, follow these steps:

1. From the Mail menu, choose Preferences.
2. In the Preferences window, click the Accounts tab if it is not already selected.
3. In the Accounts list, select your iCloud email address.
4. Click the Account Information tab.
5. Verify your SMTP server settings with the following information:
   • Incoming Mail Server: imap.mail.me.com
   • User Name: Your iCloud email address
   • Password: Your iCloud password
6. Click the Advanced tab and verify the following additional settings:
   • Port: 993
   • Use Secure Sockets Layer (SSL): Should be enabled
   • Authentication: Password

For what it's worh, I noticed someone else is now having the same problem here: https://discussions.apple.com/thread/4972783?tstart=0.  It's possible there are server issues right now that have not yet been reflected on the iCloud status website (http://www.apple.com/asia/support/systemstatus/).  You might just want to wait a while to see if it clears up on its own.

Open "Internet Accounts" in Preferences, click on iCloud to open your account preferences, deselect the mail option, and then reselect it a few seconds later and allow it to reset the settings and... Viola.

At least this is what worked for me.

-KB

I now have the same issue, but only on my mac, iPHone and iPad are working fine. Tried everything suggested here with no luck. Has anyone solved this yet. All my settings are correct, and it still says there is a server problem.

I had the same problem. The only thing that resolved this issue for me was to delete my icloud account from my mail server. (mail, preferences, click on icloud account and "-" button)

You have to click "Ok" several times if you have contacts, calendars etc added to you icloud account. Don't worry about this - they will all come back once you reactivate. Completely delete the account and then click "+" to add icloud back onto your computer.

This immediately resolved the issue of not being able to connect to the server.

Good luck!