If I understand your post correctly you have one cable line into the house, and all the PC’s talk to the world through that one line (via various boxes). In that case all the PC’s will appear to the server as having the same IP. They have different IP’s on your network but those IP’s aren’t seen on the other side of the router, the server sees the IP assigned by the cable co, that’s the one you will see if you browse to www.whatismyip.com from any of the PC’s.
I agree, normally I would expect a “too many connections” error, but it was worth a thought.