Short answer, yes. And it happens frequently that some network is cracked through a game. Of course, it depends on the skill and persistence of the attacker and that in turn depends somewhat on if they perceive some value in the rest of the system they can get to.
I don't play games much myself. We had a decent network here (about 10 machines) that we used for software development, mostly windows. We had our inet access and firewall done so tight we never contracted a virus even so, though we captured a few for analysis that came in on email. We spent considerable time setting up a good firewall, closing all unused ports on it and all machines, using non routing addresses - all the basic stuff you did back then.
Then one day we decided to play Civilization (I like version II as about the only computer game I'll spend any time on). We set it up multiplayer inside our LAN.
In seconds, we had 10 more people wanting to join the game - they'd figured out some way to tunnel comm out to the 'net even through a firewall set up only for web browsing and email (and no, we didn't even use the standard port numbers).
So, any vulnerability in that game comm software that could be hacked would pretty much have handed a hacker our entire network, since inside it, machines were set to trust one another (why have a network if things can't talk?). Since we couldn't get access to the code (of course, that was their company jewels) we couldn't evaluate if there were issues with the security of it, but the quality of the game software was such that there were worries.
We pulled the plug on the 'net for that. There was no easy way, even with wireshark, to really figure out what our risk was, but if your network has things on it like "coporate jewel source code" - and ours had among other things, full source for windows, all Ti's software dev tools, microchips upcoming DsPIC design, and a few other choice items....you get real careful real quick.
Bruce Schneier's blog is a good place to look for information on this - most of this sort of practical stuff is on there but you have to search way back, they mostly talk more esoteric security issues and philosophy now.
http://www.schneier.com/