connections to server "p59-imap.mail.me.com" on default server timed out what's the solution?

Hello Cardiac Medicine,

I understand you've been getting a message in Mail on your Mac that says connections to the server failed. I know how important it is for Mail to work properly. I'd like to help you get this resolved.

Based off of the screenshot you provided it appears this may be related to incoming mail on your @me email account. Is that the type of email account you're experiencing issues with? Does this message randomly appear, or does it only appear when you're doing something specific in Mail (e.g., sending a new message, selecting a specific mailbox, etc.)? Try out any relevant troubleshooting steps listed in the following suport article, including the steps in the "First things first" section: Get help using iCloud Mail. Respond back if you need continued support with this.

Thanks for using Apple Support Communities.

Take care.