If you read about the DHCP protocol, you'll encounter the term "lease". The IP given to a system by DHCP is typically leased, and can be changed when that lease expires. If the end system refuses to change IP when the lease expires, you'll have an IP conflict on that network. In the case of your router ignoring an ISP's DHCP lease ending, you'll lose connectivity.
It may be that when your lease expires that your ISP is giving you the same one again if no one else has claimed that IP address in the meantime. You can't count on that always being the case, though. You never know when they may reorganize their network and give you a different one.
If your lease expires and you weren't supposed to get the same IP via DHCP, it means someone else in the cable company's network has it. So traffic going to what used to be your IP will be going to someone else.
There is no rule, law or limitation saying that the ISP has to allow current connections to complete or expire before switching IP addresses on you.
If you want a static ip address you should pay your internet provider for a static ip address. Eventually the client will close the connection and that's when your dynamic ip address would change. In other words the connection likely would be forced closed when the lease expires. – Ramhound – 2013-03-07T09:20:41.867
OK I thought they would maybe do something like that. – Phoenix Logan – 2013-03-07T19:10:45.710