I assume that you have any version of Windows.
Most likely yes, you're IP will change almost every single time that you open your computer and move your hardware around. Even if you simply open your tower, remove your network card and re-install it in the same slot, Windows will say that new hardware has found and that needs to be installed. Of course, it will find the drivers because they were installed previously.
Even more true if you change your motherboard.
After that, your DNS host will assign you a new IP to connect to the network, and more often than not, it will be different.
If you remember your old ip, you can check your new ip address by going to the command window (start > Execute> cmd) and writing ipconfig /all and hitting enter.
It will give you all your networking configuration: current ip address, subnetwork mask, DNS servers, etc.
But that's in your local network, though.
If you're talking about your public IP address (which I suspect you do) and you never turned off your router or cable-modem, you should have the same... although from time to time, ISP providers renew public IP addresses for all their customers and change them.