Mail unable to verify account name or password

It might help if you could say more about the circumstances in which you are running into this problem. I have had this happen numerous times with various versions of Mail and MacOS. Often, the solution has been to wait a while and try again. The issue may depend on what email service you are using, so more info would be useful.

First, open Mail. Next, click on the Mail heading at the top to open that menu. Now click on Settings. In the Settings panel that opens, click on one of the accounts you are having trouble with. Now, click on Server Settings in the Account window. If "Automatically manage connection settings" is checked, uncheck that for both Incoming Mail Server and Outgoing Mail Server. Re-enter you account password for both servers. Also be sure your User Names and Server Host names are correct. For most servers, Authentication should be set to Password. Click Save if that is needed, then close this panel and try downloading mail.

Give this a try: boot into Safe Mode according to How to use safe mode on your Mac and test to see if the problem persists. Reboot normally and test again.

NOTE 1: Safe Mode boot can take up to 3 - 5 minutes as it's doing the following; 

• Verifies your startup disk and attempts to repair directory issues, if needed

• Loads only required kernel extensions (prevents 3rd party kernel/extensions from loading)

• Prevents Startup Items and Login Items from opening automatically

• Disables user-installed fonts 

• Deletes font caches, kernel cache, and other system cache files

NOTE 2: if you have a wireless keyboard with rechargeable batteries connect it with its charging cable before booting into Safe Mode. This makes it act as a wired keyboard as will insure a successful boot into Safe Mode.

Much like Apple

When using a Non Apple email client ( application ) to access Apple iCloud email

You may need to create a Application Specific Password in live.co.uk and outlook.com

You would require using your web browser and log into their web portal

You may need to do similar for live.co.uk and outlook.com

Sign in to apps with your Apple ID using app-specific passwords

You could always try removing the email and re-entering it. If it is a Hotmail or Outlook email address udse the Exchange option when setting it up.

It may be that there is some port setting or other item that the server you are trying to connect with is refusing. My suggestion now is to go back into the Accounts>Server Settings pane. I previously had you uncheck the "Automatically manage connection settings" items. Now, I suggest checking these so that Mail can query the server on what more it needs.

Have you installed and run any "cleaning", "optimizing", "speed-up", anti-virus or VPN apps on your Mac?