A couple of other things I meant to mention in earlier posts:
1. You said that your network cable is known working, but you didn't mention how you determined that. The reason I mention this is because you might be seeing a straight-through vs. crossover cable issue depending on exactly what networking components you are using and how the networked elements are connected.
Nowadays, it is common for networking equipment to have auto mdi/mdix determination which means that you can get away with connecting two arbitrary network devices with either a straight-through cable or a crossover cable. The network equipment itself determines the cable configuration and adjusts accordingly. But in the days prior to auto mdi/mdix, you had to use the correct cable type (straight-through vs. crossover) in a given circumstance in order for the components to properly connect. Generally speaking, like-type devices had to have a crossover cable, and you would generally use a straight-through type cable in other circumstances.
Let's say you got your TP network cable from some third-party who didn't tell you whether it is a crossover cable or a straight-through cable. Let's also say you tested your TP network cable in a comparatively new network where the hubs/routers/switches had auto mdi/mdix, and the cable seemed to work. In this case, the cable seems to work, but that doesn't really tell you whether the cable is a crossover cable or a straight-through cable. Now, let's say you try to get your ancient 3COM adapter to work in another network with older hubs/switches/routers that don't support auto mdi/mdix. Unfortunately, it now matters whether your cable is crossover or straight-through, and even through you successfully tested your TP network cable, and even though your 3COM NIC diagnostics checked out fine, you may not be able to get the 3COM adapter to communicate with the rest of the network if you are using a crossover cable when you should be using a straight-through cable or vice versa.
2. If you haven't already done it, you probably want to make sure that the Link lights on your /hubs/switches/router indicate a successful connection at the intended speed. Some of the equipment that I have used had a different color light to reflect the type of connection (10 Mbps vs. 100 Mbps.) and a solid light indicated that a link had been established while a blinking light indicates data transfer is in progress or a link negotiation was underway. Your 3COM card has separate indicators for link and data transmission, so it should be easy to confirm that a link has been established from the card's standpoint.
3. Sometimes individual ports in a hub/switch/router die or malfunction, so make sure to try a different port if things are not working. If the networking equipment is pre-auto mdi/mdix, then you need to make sure you are connecting to an appropriate port on the hub/switch/router with the correct cable type (crossover vs. straight-through).