First, you should not be giving cp the -R flag if you are intending to copy files rather than directories.

Second, now we know that we are looking at a Macintosh... Classic MacOS files had both data and resource forks. In MacOS X, most of the system plays games where a "file" can actually be a directory in the underlying *nix environment to emulate the classic resource fork. Long story short, are you sure that the "files" on your desktop are actually files?