Ill tell you about my Cox service. Since I've had it, I've had the same IP. I think that the dhcp server on my node establishes a lease for an ip on my modem. If I disconnected for a week or more, I bet I would lose my lease on that particular IP and get a new, different one. otherwise, everytime I connect, I get the same IP. Even when releasing and renewing, same ip. I suppose you could have a cable tech change your IP by telneting into the router that controls DHCP for you. As long as the tech is on a higher tier than your regular Joe at the 1-800 number, they might know what your talking about and be able to do something about it.
By the way, alot of those messages about your ip being recorded for using a bad password from the links board are fulla crap. they aren't going to do any legal action or anything. If they did, I would have so many lawsuits... But I don't.