Update: I got my mechanically-inclined DH to look at it when he got home from work a little bit ago. There's a little hook that comes out, grabs the threads, and somehow pulls them to the cutter (which we can't actually see). The hook keeps getting stuck in the "out" position. It did that once before, a long time ago. Turning the machine off, then back on, makes the hook retract. However, the next time I push the cutter button, the hook stays "out" again.
It could just be a coincidence, but the hook reaches directly toward the broken area in the bobbin case, so it's possible that the problem lies there. I ordered a new bobbin case a little bit ago. Should be here next week.
It's now cutting the top thread, and isn't making as unhappy of a sound as it previously was. It still doesn't sound quite right, though, and the bobbin thread isn't being cut. The bobbin thread is staying in the hook when it's in the "out" position.
The machine is just expensive enough (paid $399) to not be considered "disposable", but not so expensive that I can justify a big repair bill. Kind of frustrating! I'm sure it's out of warranty since I've had it almost 3 years now. The only local authorized Brother service center is a business I don't care to deal with (they were horribly rude to DH). The next-closest is 70+ miles away. I'm not sure if there are any local "un-authorized" repair shops that'll work on it.
With a little luck, the new bobbin case may fix it. >>crosses fingers