I'm not sure if it's actually a race; it probably depends on how bash is implemented. But this prints what is expected:

echo "1\n2\n3\n4\n" > numbers && cat numbers | { sleep 1 && cat > numbers && cat numbers ; }

That's not exactly the same (because the redirection for cat #2 now happens in a subshell), but it shows that this is basically racy. :)