Only reason this could be is if your POP3 server is retaining (or restoring) the messages.
Normally once your OE has downloaded the mail, it's deleted from the server. You can configure it to leave it on the server, in which case it keeps track of the UIDL of the last downloaded message so that it knows which ones it already has, and only downloads new ones. If you have it configured this way and the UIDL is being lost, that would cause it to download all messages again.
So, the first question is do you have OE configured to leave a copy of the email on the server? If you do, and it's by intention (perhaps because you want any or all three workstations to get all emails) you might want to consider using an IMAP server instead.