Same problem here in Bergen, Norway. Mac Pro early 2009, El Capitan 10.11.6, Apple Mail 9.3 from 2016. Called Apple Support and suddenly - 20 min. later it is working again
Out of the blue my Mac Mail on my Mac Pro 2008 running El Capitan is not allowing incoming email with the following error....
"Account Error > ALERT: Unable to Connect
There may be a problem with the mail server or network. Verify the settings for account “iCloud” or try again.
The server returned the error: Connections to the server “p71-imap.mail.me.com” on the default ports timed out."
I have tried deleting my iCloud account and adding it back on. I know the password is correct as all the other icloud apps are working and updating eg. Notes for example
I have also tried shutting down, pulling out the power, turning back on with Option + Command + P + R to reset the PRAM. But still the same error. Any ideas how to fix this?