Exploding Garrmondo Weiner Interactive Swiss Army Penis - (Simple?) Network troubleshooting.

My internet works. Just want to get that out of the way.

Code:

Cable -> router -> Computer 1 (internet works fine)

           |

           V

      Computer 2 (internet suddenly doesn't work)

When I ping router from Computer 2, I get "Host unreachable." It could be a problem with the RJ cable to the router and it could be a problem with a faulty network card. The card was installed a few years back and is somewhat loose.

How do I pinpoint the problem to one of these two? Or something else, really.
Btw, the cable is long and span three stories - I am rather uninclined to move it or attach it to another computer :(