If Postbox keeps downloading the same messages for a POP account the popstate.dat file may be corrupted. That's a file in the profile that keeps track of what messages have been downloaded from that POP3 server.
- Exit Postbox.
- Find your profile and then the Mail subdirectory.
- Go to the subdirectory named after the mail server used by the account you're having problems with and delete the popstate.dat file. For example if you're using gmail it would be in Mail\pop.gmail.com.
- Start Postbox and check for new mail. It will download all of the messages for that account one last time since it has no record of having ever downloaded them. However, after that it should work normally.