The web is full of all kinds of advice if you find yourself with a ThinkPad running Windows 7 that refuses to charge. The message "plugged in, not charging" being what you see in the tool tip if you hover over the battery icon in the system tray. (The Windows battery icon, that is, not the Lenovo one next door. I love the way Lenovo ship their own battery and wifi widgets, so you get two of each, and I really love the way that -- unlike the Microsoft ones -- the Lenovo ones get stretched and blurry at screen resolutions that Lenovo ship. Nice touch. With attention to detail like that from their competitors, it's no wonder Apple has such a hard time in the laptop market.)
Anyway, none of the advice I saw on the web helped (though all the BIOS updating and other fun was hopefully not completely wasted). The trick turned out to be that there's a little catch on the underside of the laptop, next to the battery. When you push the battery into place, that catch goes from "unlocked" to somewhere in the middle, just by the mechanical force of battery insertion. But you actually have to physically move the catch all the way over to the "locked" side before you can charge. There's nothing in the UI to indicate this, not even if you run the system tests.
Like I said: with attention to detail like that from their competitors, it's no wonder Apple has such a hard time in the laptop market.