This happened when a users password was changed from the command line using the 'passwd' command.
I don't know all of the details of why using the 'passwd' command to change a password messes things up, but it's all explained here www.virtualmin.com/node/13499
Along those lines, here's how I got rid of the "Usermin Failed to login to IMAP server : [AUTHENTICATIONFAILED] Authentication failed." error.
I just logged in to Usermin. Clicked on "Usermin", then on "Login", and finally on "Change Password" where it says "Use this page to change the password you use for logging into the server via SSH, telnet or FTP, or for picking up email via POP3."
I changed my password using the form and then I clicked on "Mail" at the upper left corner of Usermin and I was now able to log in and view my Inbox.