Thanks, Mac_Master. I am told by my server (a local Vermont one) that all my archives for that account would be deleted if I try your method. (I download e-mails onto my desktop computer.) I do have backup, but I'm scouting around for other solutions.
The server guy suggested that the problem is b/c of having multiple devices checking the same account for new mail. (When I called AppleCare, they said that should not be a problem, so it appears they may be wrong.) I do have an iPad, and it was set to check for mail manually, but I just discovered another layer of settings (under Settings - Fetch New Data - Manually - Advanced - ANOTHER place to check Manually. The guy said a device was checking every 5 minutes throughout the night (when my desktop computer is shut down but iPad is usually in Sleep). I seem to have had the problem even before getting my iPad -- but we'll see if this clears it up.