Hey UnkleE! I’ve been there, Hope this helps. Since it’s BOTH accounts, there could be an issue either with the network, (As you said their isn’t), the settings, configuration, credentials, or an OS issue.
Are you able to sign into your email provider’s web site? If so, you can try these steps, if not your Mac might require an updated password:
First, before removing an account from Internet Accounts, I export any mailboxes I need as a backup, such as the inbox:
https://support.apple.com/guide/mail/import-or-export-mailboxes-mlhlp1030/mac
Select the Mailbox and select Mailbox > Export Mailbox, and choose a destination. Later, if you are missing any mail, you can choose File > Import Mailboxes, as in the article.
This is especially important for the mailboxes that are on the “On My Mac” section as those don’t sync with your email provider.
Typically, if your email account is considered IMAP, the Mail is synced and stored on the email providers server and you can safely delete the account and then re add it.
If it’s a POP account, deleting it will most likely delete all of your emails, so it’s always good to have a backup, either way I’d export for a copy, Time Machine also does backup any local mail, (As in On My Mac), so that’s always a good idea.
So if you’re connected to a network that you know is working, and even on another network that is good, I’d try these steps:
The path of least resistance is to select Mailbox > Take Accounts Online, (It May ask for your email password).
https://support.apple.com/guide/mail/take-email-accounts-online-or-offline-mlhlp1032/mac
Also, if you open Mail, on the top menu click Mail > Preferences > Accounts, select the account not working, then in “Server Settings”, and make sure it’s not requesting your password and that all the settings are correct, then click “Save”, if available.
Most email settings can be found on Apple’s web site:
https:https://support.apple.com/en-us/HT202480
However mail providers change these settings often, so if you need to, contact them for the current settings.
Chances are though, if it happened on both accounts at the same time, there maybe something else going on.
Are any other internet services not working? As in loading a web page, etc?
Either way, it may be a good idea to check for any type of malware/adware that may have been installed unintentionally.
Updating the OS and restarting is a good idea as it initiated the built in Malware Removal Tool, on your Mac,
Malware Bytes for Mac is another good resource for finding any remaining type of malicious software:
https://www.malwarebytes.com/mac/
If you’re still having issues, changing the network location is a good step:
https://support.apple.com/en-us/HT202480
Also, If you see anything in System Prefrences > Profiles, that you didn’t install, with a name you don’t recognize, I’d remove it by highlighting it and pressing the “-“ sign below.
Another place to check is also in the System Preferences > Network > Advanced > Proxies, if you see something there, other than the bottom option checked that you didn’t do yourself, I’d in select it, hit OK then “Apply”, then restart to test.
Sometimes in the same network settings, if you’ve installed some networking or security software, you’ll see an extra interface in System Preferences > Network, in the left column, if you need to, or don’t use the program anymore, I’d remove it from there, (Again using the minus sign), then restart.
(Safe boot is a good way to test if third party software may be causing an issue):
https://support.apple.com/en-gb/HT201262
Another thing you might want to check is your login items, which are applications that open automatically every time you login. Chances maybe that the app that’s interfering with mail may be in here, you can also remove them from System Preferences > Users and Groups > Login Items, highlighting and also pressing the minus sign to remove, them restarting. (This doesn’t remove the program, it just prevents it from starting up at every login. You can add or remove programs to this list at any time.
Another thing to add, is many email providers are updating their security measures, including, requiring “App Specific Passwords”, to be able to log into a particular device in order for that device to access that mail. So you can check with your email provider about that.
For instance, Google, (Gmail), recently updated their security requiring Mac and Apple users to allow access to your Google Mail. If it didn’t prompt you automatically, then sometimes, yes, deleting the account and re-adding it will do the trick, allowing you to allow your Mac to access your Gmail mail. Other providers are doing similar things.
Hope that helps, and hope it’s a simple fix!
