Not always, it's up to your ISP to assign your IP address to the central device on your connection (usually your router). From there your router gives computers a local IP address that is just for your network. Most ISPs will go out of their way to reassign the same IP to a modem. With comcast, I won't get a new IP even if I unplug for 5 or 6 days usually (I unplug everything when I travel).
If the two laptops (one working and one not) are on the same network, they will have the same public ip from your ISP; if one works and one doesn't, then your IP isn't how they blocked you. Other methods, as stated above are by MAC address of your laptop, by your account, or by a file on your computer.
As others suggested, try calling the company first. They want to protect against cheating, but their records should be able to show you weren't playing two accounts at once (assuming you were not), and a human maybe able to make a determination to restore your access without working around their system (which could get you more trouble).