It's possible if your mail goes to a user who's got shell/ftp access, it's not possible on a user account which just has mail access.
FTP to the server with the username and password of the user who has received the mail. You've got a /home/_username_/Maildir directory which contains all the messages. To import those messages into your local mail program (like Thunderbird) you'll most likely need the 'mbox' format and not the Maildir format. You can get scripts that convert Maildir to mbox here: http://www.systemsaligned.com/content/view/18/19/ (also contains instructions). You'll need to ssh into you DH account and runs the scripts on the DH server. I have never tried this, but google turned up this solution.
Another idea: give that person who is sending you the huge attachments (DH only allows 45MB attachments anyway) a FTP-only account and let them upload the files to it. This would be the easiest solution.
Hope this helps.
Promo-Code: [color=#CC0000]SAVEMONEY97[/color] - Save [color=#CC0000]$97[/color] on your first year of hosting.
Use [color=#CC0000]SAVE2DOMAIN[/color] to save [color=#CC0000]$82[/color] and get a second domain for free