Let me clear up the conflicting answers you have received.
Yes. The IP will change. Pretty much every ISP works with Dynamic IP's.. Cable or DSL.
However, generally they give very long lease times -- weeks, months, or longer. So even though the IP is Dynamic (changing) it often stays the same for a very long time making it appear to be Static.
The ISP issues IP's based on MAC addresses of devices -- This is how the head-ends work at cable companies.. I won't get into the networking of it all, but just suffice it to say when a MAC address changes of the device the ISP's headend talks to, the IP is re-issued and the IP will NOT be the same as the previous one.
Every modem has a unique MAC address -- so a new modem (for example in Cable systems) -- will send data through the local node, to the headend to request service based on it's provisioning. The system will acquire Quadrature Amplitude Modulation (generally) it's downstream sync and then send T3 acquisition requests for the Upstream based on the frequency and then negotiation is done between the two to establish the best signal at the cleanest "power level" in dB -- The higher the power level the more the headend has to 'amplify' the signal to send clean data reliably to the modem and for the modem to send data reliably back to the headend. But I am getting into the weeds here.
A new MAC from a modem = New issue of IP from the available Pool of free DHCP controlled IP's. Thus the IP will change.
For future reference it is , especially with cable systems, not too hard to cause a change in IP. You can cause the router to Clone the PC's network adapter's MAC address and then release and reacquire the IP -- this will usually be enough to get a new IP issued. Then to switch again, you unclone and let the router use it's own MAC (as it comes from the factory) and another issue of IP occurs.
Further, if you unplug a cable modem and then plug it back in most of the time the Same IP will be given.. this is due to 'preferred' IP common with the router at the headend -- recognizing the recent MAC and giving it the same IP assignment that is has not been re-routed. However, if you unplug and let it stay unplugged for a day or longer.. more times than not the IP pool will have changed and the Modem will be issued another IP. This relies a bit on timing and luck.. but it works.
With DSL modems the routing is a bit different, but all routing relies on the same TCP/IP principles, and network ranges.
Further a new modem will need to be provisioned -- this process alone will cause a different IP to be assigned.
So, yes, if you or a friend are banned at the IP level.. then this will alleviate such a ban. If you are banned on a User account level then it will not.
Take care,