No, I could ping outside the network...
In the end I figured it out. The MAC assigned to every other network card in my house was in the 00:xx:xx:xx:xx:xx range, whereas my nForce card was in the BA:xx:xx:xx:xx:xx range. I'm guessing my ethernet switch doesn't like big MAC addesses, because when I changed to something lower, everything started working fine.
I'm guessing the reason I could ping and resolve DNS is because these are both connectionless protocols (ping over ICMP, and DNS over UDP). Interestingly, when I tried installing Windows XP, neither ping nor DNS would work.
All very strange. But at least it works now. :P