Are the delivery notifications on your cell received as emails or text messages? If they are text messages, you'll want to check with Verizon regarding bounced email notification preferences, something they control based on your account settings.

On the other hand, if they are emails, I would:
1.) Check if your are using IMAP or POP3 through both cell and desktop email clients.
2.) Check the settings of both email clients to ensure that the 'leave a copy on the server' option is selected, if applicable.
3.) See if other inbound email is similarly affected.

John