There could be a few ways of doing this. If you are ok with SD composite inputs you could start with a kramer matrix switcher, this means you can send the same camera to multiple computers you can have 4 capture inputs on each machine (not so easy but it can be done). This way the limitation is the 4 inputs at once, but if you want to have 6 inputs on every machine at once (and have all 6 inputs displayed you will have some trouble, not just because of capture cards but because this is a hell of a lot to ask of one computer- no matter how good it is.
If resolution is not important (if you are displaying all 6 cameras at once the destination resolution cannot be very big) then you can also use quad splitters (hardware) and chop pixels (actor) to get 4 inputs into one capture card.
A little more information we can help you more- a matrix will let you send each camera to many places but how many camera do you need to display at once.
I also made a ps3eye to syphon app that supports multiple cameras, you will have to have multiple pcie-usb cards in a tower machine (a sub hub won't do it) to get enough usb bandwidth to have 4 cameras going. I have had good luck with 20m USb extensions and the ps3eye cameras.
If you have money you can use SDI based cameras and an HDSDI quad split (you get almost full SD res from a single quadrant of HD), from here you can go into a blackmagic dual HDSDI card- you could have 4 quad splitters per machine and get 8 inputs on each if you needed. This would be a pretty heavy load on a single machine even 2 streams of full HD is a lot of work then the chopping to get the quadrants to a single feed but it is a more professional solution.