Just in case you actually have BNC NICs, I'd better not leave you with the idea that you can just connect the cable directly between two of them. You have to use a T-connector on each NIC, and run the cable between the T-connectors. If you connect the coax directly to the NIC, it won't work. Might work just long enough to make you THINK it's working... but as soon as you start to depend on it, it'll break your heart. Trust me.
On the open leg of the T, you must have a terminator plug. To add a third computer, remove the terminator, run another cable from the connector thus exposed, and put the terminator on the new computer's T-connector. It looks like this:
Note that there is always a terminator on each end.
Because although it looks pretty darn thick, it's still thinner than what it replaced. Simple as that.
And in the "been there, done that" category -- no, you can't use TV coax in place of networking coax. Even if you replace the connectors, the cable's own resistance is different, and you'll build up alarming voltages in places that aren't supposed to have them. If you get a shock when you're trying to hook up your network, something is dreadfully wrong.