I have a single user who receives the "New mail" icon in the system tray, but outlook hasn't downloaded any new mail when she looks in her "inbox" or any other folder.
Is she getting the dekstop alert toast as well? If not, then the icon notification may be messed up. You may want to simply turn it off. Do the reverse of this: http://support.microsoft.com/kb/292034 How to restore the Envelope icon in the notification area in Outlook