Anytime you add a computer to your personal network, you're essentially dividing your bandwidth. However you shouldn't have much problem with lag as long as each component you're using to connect to the internet is fast and reliable. You're only as fast as your slowest part.
My suggestion would be to make sure that you buy a name-brand, high quality router (not a switch). Each of the big networking hardware manufacturers have entry-level devices. Avoid those and go straight to the middle/top-line models.
Also, although unnecessary, it might not hurt to get a dedicated NIC in each of the three computers.
Finally, if you're renting a Cable modem from your service provider, return it and buy your own. That way you can use a high quality modem as well. If you've taken all the above steps (and providing your computers themselves aren't lagging) you should be good to host more than three computers for internet gaming.