The problem persisted through 22.214.171.124, and disabling all extensions didn't seem to help. I did find that toggling "Work Offline" would clear things up for a little bit, and I used it as a workaround for a while. But that got old pretty quick.
I took a look at my about:config network settings, and stumbled on to the fact that my network.http.max-connections had set itself to 74. I moved it back to a more reasonable number (48), and that fixed it. Regular browsing resumed, and I disabled FasterFox, which I figured (incorrectly) was causing the trouble.
The problem started recurring intermittently. Since I knew what to look for, it was easy to fix, but I did a little more looking and finally found that the problem was coming from DownThemAll (not "down the mall"), and that it was a known issue. So now I know. And knowing is approximately 50 percent of the battle.
DownThemAll is a stunningly useful Firefox extension, in that it makes batch downloading of files linked from Web pages really easy. So, if you're looking at a page that has lots of, say, image thumbnails that link off to source images (alright, I'll say it: porn. DownThemAll is really good for downloading porn), DownThemAll is a quick way to snag them all.
I'm sure it's also useful to other photography enthusiasts and such.