Most email clients remove the mail from the mail server, so if you use Outlook on your desktop to check your mail, then you won't see those messages on your eDGe anymore. From your description, you have things set up to remove messages from the server. Note that at least one of your email clients has to remove the messages from the server, otherwise your email box will fill up and the server will bounce back all messages after that until you empty your email box.

My laptop is my primary email client, so I set it to download messages, removing them from the server. On my eDGes (I ahve both the EE and PE), I connect to the same accounts but in the app settings I specify that emails should be left on the server. When I'm away from home I shutdown my laptop at home and then I get the emails on my eDGe. When I return home and fire up Outlook, it retrieves all the messages (and after that those messages no longer appear on my eDGe).