the connection to the server "mail.mac.com" on port 993 timed out

I'd get a backup of your disk and you rmail before commencing here.

If you're using MobileMe and haven't migrated to iCloud, then I'd expect to use smtp.me.com and mail.me.com for sending and receiving mail; you're apparently on the older settings from the .Mac stuff; here are the MobileMe settings (HT2296) needed.

If you've migrated to iCloud, then use the iCloud mail entry in your Apple Mail set-up, if you have that. If you don't have that iClound entry in your mail client, then you'll be entering and using theiCloud settings (HT4864) directly.

I don't know if the old MobileMe settings will continue past the end of the migration to iCloud. Apple states: "MobileMe ends soon. Move to iCloud before June 30, 2012, to keep your mail, contacts, and calendars up to date automatically and get Photo Stream, Backup, Documents in the Cloud, Find My Friends, and more. Learn more about moving to iCloud."

And AFAIK, iCloud is not available on 10.5.8, if the version shown in your tag line is correct.