Open Network Utility and select your Ethernet adapter. It should tell you what's going on with the card itself with the "Info" button selected. If it shows information like an IP address, then something really odd is going on. More likely it will show no IP address, but it should show a hardware address, indicating that the card is operational.
The next thing to do is to get a small toothbrush or something similar, and after turning off the computer and unplugging the cable, scrub that Ethernet port! Over time, dust and humidity can cause problems in this sort of connector, and while the connector is similar to phone jacks, it's bigger and can thus trap more gunk. We also use computers in places that get less air circulation—and can thus collect more humidity— than phones, so this can also contribute to the problem.
It really looks like a hardware problem, and if it's not the connector (and you've proven it's not the cable), then it may be that the Ethernet adapter has failed.