It is possible, with the use of sticky bit. If you set it on a directory, then files inside that directory can only be renamed or removed by the owner of that directory, even if other users have write permissions to the files there.
chmod +t /home/username/dirname
That should do it. If you'll look at the permissions of a directory where sticky bit is set, you'll notice there are "t" letters where the usual "x" (execute) letters usually are. See
for an example