Huh. Changing browsers seems to fix it. I'd been using Safari, but once I tried Firefox the problem went away. Thanks for the idea.
But I'm still concerned. I've been using Safari as my primary browser for close to a year and I never had anything like this problem before. For the record, it was specifically VF and I Should Be Writing that gave me trouble; I had no problem downloading NPR podcasts, BBC podcasts, Drabblecast, anything put out by Escape Artists, or a couple dozen other miscellaneous mp3s I've grabbed from around the Internet. Weird.