Sorry, in your last post you said you didn't loose your IP number from your router, just your connection. Now you say a Dell laptop also says it has a limited connection, which could mean the same thing. You can be getting a LAN IP number from the router, but the router doesn't have a WAN IP # so that's limited connectivity.
Since both machines lose the connection I don't think you can renew it by hitting the "button" in network prefs on the mac.
The router doesn't seem to be the problem, because you were having connectivity issues before you started using it. Everything points to the cable modem dropping the connection for some reason.
You never mentioned using two different computers on the connection before, so I assumed you wouldn't be running into a MAC number problem on the modem. That may be the problem, as your router does not have the same MAC number as the computer's ethernet card. The router usually has a setting where you can spoof the MAC setting for the computer into the router so that it identifies itself the same as the machine that used to connect to it. Not too many people have to actually spoof a MAC address anymore, so I'm not sure if that's it, or if the modem is just crapping out on you.
Now that you bring the Dell computer into the equation, and have added a router, and the nature of the problem has changed from a phantom IP number to a connection failure problem I would start with replacing the cable modem.