Against casual attackers, NAT acts as a firewall with a "default deny" policy for inbound connections and "default permit" for outbound connections. So you can use this setup by placing a second router in your network for, for example, wireless guest access.
This is a very specific scenario (one which I actually use for one of my customers). His claim seem to be more general, as to the fact that having two layers of NAT would somehow improve security. I can't see any reason why it would, it would only create a lot of headaches with applications that are already struggling to work properly through a single layer of NAT.