I have one of those Linksys routers at home. I can open up a single port at 47624, but I think the max is 10 redirected ports.
If the 23000-24000 ports are initiated my my machine, they can pass through the firewall both ways. But if the host game is trying to connect to me at a random port (well, not random--rather negotiated by some "master" connection ahead of time), I'm out of luck. It would be too bad if that were the case. I don't know of any practical use for this design (ftp in passive mode does this, but I don't know what advantage that has besides historical compatibility) and it would lock out a significant number of people.
The host machine may need more ports open. I'm just thinking of the client machines connecting to it.