I had this problem today for the first time. I have a Time Warner Road Runner account and when Apple told me that the emails were not being deleted from the server, I called Road Runner. I was informed that they were "upgrading " their system and "backed up" users emails so that nothing would be deleted when they made changes.
I was directed to go to the Road Runner mail account
https://webmail.nyc.rr.com
There you may check to see which emails have been read and not and deleting those not needed will remove them from the server.
Also I should add that I went to Outlook Express and changed my setting to remove emails from server after 1 day (had it set for 5 days.).( Go to Outlook Express - select Tools-select Accounts- select Properties- select Advanced- select remove from server after 1 day- select Apply.)
So far so good. But tomorrow will be a better test.
By the way, they told that the upgrade was not completed so I'm expecting the worst.
Hope this helps