If you need to bind mount some directory in read-only mode (to securely access a backup dir, for example) it is necessary to first bind mount and then remount in read-only mode.
At least on Fedora, you can perform these two steps automatically using /etc/fstab. Just create a pair of entries like these:
/source/dir /destination/dir none bind 0 0 /source/dir /destination/dir none remount,bind,ro 0 0
I've tested this in a Fedora 14 box. YMMV.
Update: This trick only works if your distribution does a mount -a during boot. If your distribution uses systemd, you can try this other hack.