I did a ton of google, wiki and forum searching to look for a solution, but all I found were tons of reports of the same thing with no known solution. I then recalled that I had installed the MADFilter codec (madfilter.ax) to fix radio stuttering issues in Fallout 3; out of desperation, I used InstalledCodec to disable MADFilter, and that seems to have fixed the problem (I've been listening to Radio New Vegas for the better part of an hour with no song dropouts).
Ironic that the fix for one Fallout game causes problems in the same area for another Fallout game using the same engine... Fortunately, it's easy to disable/enable MADFilter via InstalledCodec, without needing to reboot or anything.