Not every computer has its own IP address. True or False?

At one point when I was building a voter based contest, I considered facilitating the integrity of the competition by allowing only one voter per day based on the user's IP address. I chose not to for the following reasons.

If a family is using one computer, not everybody is going to be able to vote because while that computer may have a unique IP address, not every user does.

Secondly, you have more than one "layer" of IP addresses. You have that IP address which is accessible to the public over the internet and the IP address which is unique to your box. So on a LAN network, the world sees the IP address of your router, but beyond that, they can't see the individual IP address of your computer. Your Local Area Network can, but the world cannot.

Is my understanding accurate?