It wasn't just http requests that were failing; I couldn't successfully ping, tracert, or connect via ftp to any of the "non-working" sites, either. I was able to successfully connect to some of the sites from work; plus, given that if a site like the google.com homepage had any significant downtime at all it was likely to immediately be big news, and there had been no such news, I was able to rule out the web server as the problem. I was able to reproduce the problem on a couple of different client machines and different web browsers from home, so that (most likely) ruled out the client PC as the issue. My suspicion at this point was some kind of problem at the ISP level; I figured that since I was able to browse some websites with no problem at all, my router and cable modem must be ok.
This turned out to be a bad assumption. I had tried "rebooting" the cable modem (by disconnecting the power, waiting a couple of minutes, and then reconnecting) early in the troubleshooting process, but that step had had no effect. Throughout the process, I had assumed my router was ok. However, when I used the router's browser-based administration application to try and perform a release/renew on its Internet IP address, and the application simply stopped responding after the new request, I immediately (and to my chagrin, for the first time) suspected some kind of problem with the router. I cycled the power on the router (unplug, wait briefly, plug back in), and that turned out to be the solution! Immediately, all of the 64.x.x.x sites were working fine once again.
I'm still not sure why cycling the power on the router was the solution to the issue, except that to guess that apparently, after weeks (or maybe even months) of uptime, the router had managed to get itself into a bad state, and cycling the power was enough to refresh whatever internal memory the router has back into a good state. Hopefully, this post will be a time saver for anyone else out there who manages to run across this issue -- at least, so long as this blog doesn't get changed to have a 64.x.x.x IP address! :-) I have had various problems with this router -- a Linksys BEFW11S4 Wireless-B that's now over a couple of years old -- before (1, 2, 3). I think it may be time to start keeping my eyes open for a sale, and jump on a deal to replace this router when the right deal presents itself.