I was told that CF use what they call a Hard Sticky IP's that they link up to your MAC of your routers port connected to the modem, so your connection will only work with said router unless you phone them up to clear the old MAC their system their end.
TBH I have no issues phoning them up to have a new IP allocated to my connection or to have my MAC removed from their system allowing me to swap routers.
But saying that, it would be nice if they setup a web interface for us to login and release & renew the public IP and to be able to clear the MAC.
Paul