IIS6 is refusing to host files with a space in the name. The browser requests the file as: http://mysite.com/files/this%20is%20a%20file.wmv, which is fine. But IIS just returns a 404 for it. Replacing spaces with dashes works fine. This is a new development setup, and the previous IIS6 server hosted the files just fine, so I must be missing a setting somewhere to enable this.
Just to be explicit, the MIME types for the file are set up fine, as evidenced as when I remove the spaces and uses dashes instead.