Hi Bonnie,
For email accounts configured in Outlook for Mac, their passwords will be kept in the Keychain of Mac. According to the error message, it is more likely to be related to password (password stored in the Keychain or something like that). Since deleting the password entries in the Keychain doesn’t work, you may try the following methods to give it a shot:
1. Reset default keychain.
Start the app Keychain Access
Open Preferences and select Reset My Default Keychain
2. Reset your iCloud Keychain password.
Opening iCloud under the System Preferences pane
Click Options under Keychain
Click Change Security Code and follow the instructions
And, the current newest production version of Outlook 2016 for Mac is 15.31. You can try upgrading your client via clicking “Help”> “Check for updates” and see if the issue can be resolved.
By the way, I’d like to confirm whether you are configuring the 4 accounts under the same profile of Outlook 2016 for Mac. If so, please depart them into different profiles to check the result. You may refer toOutlook 2016 for Mac profile.
Regards,
Alan