OK, if you are in Fedora Core 5, do the following:
cd /tmp
echo "text" > file
cp -p file file.1

Everything works fine. Now unmount /proc and repeat the copying:
umount /proc
cp -p file file.2

Whoops! I never thought that copying files requires the whole damn /proc partition.

I spent some reasonable time debugging that today. Will come with a bug report and (hopefully) a patch tomorrow.