I keep getting this message: Connections to the server “p44-imap.mail.me.com” on the default ports timed out. Any idea what's wrong

See these articles Use Mail Connection Doctor - Apple Support

Verify an email account’s outgoing server in Mail on Mac - Apple Support

Change Outgoing server settings in Mail on Mac - Apple Support

Automatically manage connection settings in Mail on your Mac - Apple Support

If your iCloud account is not connected ( just an example ) , open mail application click on I cloud and send a mail from - iCloud address ( in To - field type your gmail address ) the mail will be sent to your inbox in your apple mail application a pop up appears .in this screen shot it is showing could not connect to outgoing server , thats the reason when connection doctor is opened SMTP for iCloud is totally vanished .You can apply same with gmail say if IMAP or SMTP is not working . Simply click on always use this server to get joined .

And you will observe all IMAP / SMTP for gmail account and iCloud account are showing green dots and joined to their respective server .

Also if you are using a gmail account now there is no need of typing the port numbers in the fields , it will set up automatically and I am not aware which mail services you are using there is link Mail Settings Lookup - Apple Support

You type the mail address in the field and it will show your correct port settings .

Read this article Get help using iCloud Mail - Apple Support

And the issue you are facing will occur when date and time settings are changed , any read and write permissions in home folder , guest , shared folders also location services should not be altered .

Try these steps otherwise delete the IMAP account if it is used and reconfigure it Add or remove email accounts in Mail on Mac - Apple Support

All of the mails are stored in server they will start syncing , if you are using pop account then the best way is to save mails in V6 folder used in mac os Mojave 10.14 , in Mac OS high Sierra it would be V5 .

Don't use any any anti virus , third party softwares , any kind of third party app that will directly corrupt the apple inbuilt applications and system files .

The last suggestion would be one should create a new test user account if the same issue occurs do not reappear , then switch to a new user test account and transfer the data see this article How to test an issue in another user account on your Mac - Apple Support

If you have emails in your mailboxes, as a safety backup, select the mailbox, then Mail/Mailbox/Export Mailbox. If adding the account back doesn't recover your emails, you can import what was exported.

I want this article to be added please read it If you can’t send or receive email on your Mac - Apple Support

i've also been seeing this off/on for the past several days, although i have the p35 server. for me, it does not appear to be inhibiting sending or receiving mail.

i have a very good idea of what i'm doing, but thank you for posting the apple mail 101 links.

fxt

Have you tried deleting the account and adding it back?

You are welcome.

Thanks but nothing is working for me either. There are no servers listed under my mac account. It was working fine yesterday.

...forgot to add that my internet and incoming mail are OK, just can't send email and get the same message that Marios174 originally posted.

Rep who was helping me said that Apple is working on the outgoing mail SMTP issue now. (2:30 MST)