It took me close to two days to figure out why one computer couldn't access the Internet, even though it was set up identically to several machines that could, and it had no trouble talking to the internal network.
The TRENDnet TEW-633GR Wireless N Gigabit router, like many routers, has a MAC address cloning function, for sharing an Internet connection even if your ISP has tied your cablemodem to a particular network interface card. But unlike the D-Link router I was using previously, the TRENDnet router apparently does not forward packets to the computer it's pretending to be, if there's a switch between the router and the computer.
But on a positive note: Happiness is not having to daisy-chain switches any more, and still having plenty of ports open for expansion.
(Too many yellow lights on that, though; I definitely need to buy some gigabit NICs for the older machines!)