I agree with Chris, please clarify why that would be desirable!
Yes, it's possible. Instead of bash, give that user a "shell" that only understands "mv" and "cp" (note that they won't be able to log off). You could write that as a perl script. Yes, I've done it and it works; my script understood only "exit".
Beware that the user could effectively remove files even if you don't allow "rm". E.g.:
mv file-a /tmp/.gone
mv file-b /tmp/.gone
cp /dev/null /tmp/.gone