It would appear that the LinkClickHandler is properly registered in web.config. Let's look at the location of the files that are not being returned by the handler. In Admin-->File Manager:
 
1) Are the files that have been uploaded visible?
2) Into what type of folder are they uploaded - Standard, Secure, or Database Secure?
3) Have the roles which have permission to view/download the files been granted View/Read permission on that folder - for example, if both authenticated and non-authenticated users are supposed to be able to view/download have the "All Users" role been granted view/read permissions  on the folder (in Admin-->File Manager?
4) Is the host or admin user able to view/download the files without error?

Regarding my comment on upgrading, somehow I mis-read your DotNetNuke version and thought it was 4.08.02 - too late at night I guess. If you are unable to resolve this issue, I would consider an upgrade from the current 5.04.02 to 5.06.01 - again making sure to have backups of both filesystem and database.

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Bill, WESNet Designs
Team Lead - DotNetNuke Gallery Module Project (Not Actively Being Developed)
Extensions Forge Projects . . .
Current: UserExport, ContentDeJour, ePrayer, DNN NewsTicker, By Invitation
Coming Soon: FRBO-For Rent By Owner

I have direct access to the server running my website; and when this happens to me it's general because the module is calling resources from "localhost" and it's not DNS name. So it will read http://localhost/whatever.com, instead of http://mywebsite/whatever.com. Have you checked this because it will prevent all users from downloading the resource, (since it thinks the file is on the local computer and not the server) but still be able to upload to the server.