Unable to verify account name or password despite using correct details

I've had this problem for a while, but I hate not knowing or getting to the bottom of things, so here I am! :)

On an iMac I am running El Capitan 10.11.6 and in the Mail app I successfully send and receive emails from five separate accounts. So all seems good.

However, if I go to System Preferences > Internet Accounts and manually select each of these five accounts, for two of them I get the following Can't connect to the account error message:

Now, when I enter my password, I get the Unable to verify account name or password prompt in red text:

I manage my own domain and email accounts, so know for a fact I have the correct passwords. And, as I said, I am using these accounts fine within Mail. I also use them fine on various iPhones and iPads and other Macs.

Any suggestions as to what's going on?

I was having this issue also, tried restarting the computer, editing the SMTP, nothing worked. I went into internet accounts and clicked the remove button (-) in the bottom left. After removing and re-adding the account everything seems to be working fine!

This is a frustrating bug! Here's what worked for me, though I don't know which of these things ended up doing the trick (Mac OS 10.12.6, Mail 10.3):

1. Open Mail preferences and on the accounts tab paste the new password into the Server Settings tab.
2. Disable the account on the Account Information tab.
3. Restart the whole computer. (probably not necessary)
4. Launch mail, open up preferences again to the accounts tab (this time it showed a green "Online" status).
5. Re-enable the account on the account information tab.
6. At this point, for some reason Mail thinks the account is offline, though I never set it to offline. Click the get mail button and it will ask if you want to bring the account online, click bring all accounts online.

And finally after all that it started working again. This is so ridiculous. Whoever thought you would have to reboot your entire Mac just to update your email password.

I had the same problem, The solution is to delete the entry for the account password in Keychain Access.

the name for the account is:

com.apple.account.Exchange.password

there seems to be a bug as this account doesn't show till you search for a string that matches in this account, like "Exchange"

I got the hint for this solution from this answer