Now my ideas are just shooting in the dark, so do not assume I know what I'm talking about 🙂
Please try booting your Mac into "Safe" mode
<http://support.apple.com/kb/ht1564>
This will eliminate any 3rd party interference, such as an aggressive anti-virus package (which are totally NOT needed anyway). Safe mode does not load 3rd party extensions, and does not run "Startup Items", etc... So if a 3rd party package is interfering, Safe mode is an easy way to test without those packages loaded.
If you have WiFi, can you try using that to get an IP address?
Have you tried a different Ethernet cable. Maybe borrow one that you know works, such as one from the internet phone setups. While Ethernet cables do not go bad by themselves, cats, dogs, kids, humidity, or other environmental stuff might cause the cable to fail.
Is it possible to connect directly to the modem, bypassing the router (NOTE: some cable modems lock into the first device they see after being powered up, so you may need to power cycle the modem if you are going to try attaching your Mac directly to the modem, and power cycle it again, when putting the router back as the connected device).
You could try manually assigning your own IP address. I do not recommend this, unless that Mac does not leave the house (if it is a laptop that leave the house, a fixed IP address will cause problems when you attach to another network). You would look look at your router's IP addresses, and look at what the DHCP address range was on your router, and choose and address just outside the DHCP range (for example, if the router was using 192.168.1.2 thru 192.168.1.100 for its DHCP range, you could choose 192.168.1.101 (where different routers using different non-routing IP addresses; common are: 192.168.0.*, 192.168.1.*, 192.168.2.*, 10.0.0.* are the ones I've most commonly seen; you must figure out what your router uses).