My LAN cable seems to work (indicator LED lights up) but the computer can't find a connection. What's wrong?

Setup:

1. I had to run a network cable from a router in one room to a computer to another room, through a hole in the wall that was too small to pass the RJ-45 plug through. The plug was cut off and the cable passed through the wall.
2. Then a new plug was crimped on using this detailed explanation.
3. The connection didn't work because the (factory-made!) plug in the other end used a non-standard wire order.
4. I crimped a new plug on again, using the exact same order of the factory-made plug.
5. The LED indicator lights up on both ends, but the computer cannot find a connection.

What can be wrong? How can I find out?

I don't have a cable tester. By visually inspecting my new plug, I think it's good; the wire order matches the other end, and all wires are all the way inside the plug and reach the connector piece. I've used the cable before (with both factory-made ends) so i don't think that the cable itself has a defect.