This problem is so stupid and senseless that I'm not sure anyone, here or otherwise, will be able to figure out what's going on.
Background: I just moved into a new place. The house has a Netgear WNDR3400 N600 router. The router and network ostensibly work fine; all the preexisting computers in the household can connect just fine, both through ethernet and the wireless network. My laptop, which has never been brought into this house or connected to the network before, is an old-ass Acer TravelMate from 2004. Its wireless adapter and ethernet adapter are both functional.
Problem: My laptop won't connect to the network, or the router won't talk to it, whichever is the case. If I try to connect via an ethernet cable (one which I know is working), Windows isn't aware of the connection at all (the typical "network cable is unplugged" error). If I try to connect via the wireless, it hangs on "waiting for the network to respond" or whatever the first message is. It doesn't even get to DHCP in either case.
This is rather bizarre. I know the laptop's adapters can connect to other networks, and that the wireless adapter supports the encryption used here. I also know that the other computers in the house connect just fine. The router also doesn't have any MAC address filtering set up. This all makes it very hard to pin down the problem, or even identify which device is causing it.
If anyone has any ideas, go for it.