OK, so, this seems to be the solution:
I had a "double" Flash player installation. Thanks very much to Sadisticflipside for suggesting this solution! I think I have fixed the problem. I'm not 100% sure yet, since the testing will be going on for a little while just to see if bandwidth or other pressures don't screw things up again. But for now, it seems that I can stream with impunity as I always could until this problem arose.
It wasn't exactly a double-installation of Flash player. More accurately, it was one and a half installations. My method for resolving this problem was to un-install Flash, and Firefox, and a lot of other stuff, and then re-install.
And, when I say "un-install," man, I MEAN it. I not only ran the uninstaller app that comes with Flash. I ran it several times. I "forced" ran it. I manually searched and hunt-pecked for any folder named "Adobe" or "Flash" or "Shockwave" or "Player" and for any Registry entry similarly named. I deleted the ones that seemed sensible to delete. I ran CCleaner intermittently among all of these un-installations and deletions, removing more and more "problem" registration entries that it found. I ran the other computers in the house while this laptop was without Flash. I made sure things seemed OK everywhere else. Then, and only then, did I re-install (1) Firefox and then (2 ... whoops, first there was a problem.
See, this is when the weirdness arose. Get where I'm at. I have un-installed Flash, right? And I have also un-installed and then I have re-installed Firefox, right? So, that means I should have a Firefoxed but Flashless computer, right? Yes Firefox, no Flash, right? But at that point, on YouTube, I watched a buncha videos. I watched the Nye Gitarkameraten (Fuentes, Lund, Holm, Nilsen) do Leonard Cohen's "Hallelujah" on some Norwegian talk show. Then I watched K. D. Lang do the same song for the opening of the 2010 Winter Olympics in Vancouver. Then I watched K. D. Lang do the damn "Walk On" song from "South Pacific" that they sing at Liverpool football matches. Then ... I realized that there weren't any advertisements showing up on YouTube. And then I realized, that I shouldn't be able to watch YouTube if I don't have Flash on my computer, right?
So, weirdness. So, I ran the un-installer and the forced-un-installer for Flash all over again and again. I restarted. I kept running CCleaner. And finally YouTube wouldn't work right. And THEN ...
... as I was saying two paragraphs back ...
... then, and only then, did I re-install (2) Flash.
So, evidently, there were some dregs lying about. Flash had left some residue somewhere. I've gotten rid of it and finally gotten a really clean re-install of Flash, and it seems to be working fine. Unfortunately, that does mean ads have come back on YouTube. Pity ...

... but at least now it doesn't stop halfway through any stream and just sit there for five minutes (literally) while the download is not downloading.
Again, thanks very much, to Sadisticflipside, for the suggestion. I'll mention something if the solution turns out to be a false lead, but, for now at least, I'm back to simple streaming like it used to be. Also, I do wonder about two questions:
1. How did this happen? Did it have something to do with First Row Sports or NBC Sports Extra and the exorbitant high-bandwidth streaming I was doing?
2. Why do two Flash players show up as running in Task Manager? Whenever I am playing a Flash video, then Task Manager says that Flash is running. Seems sensible enough. But, it shows either NO Flash running, or TWO Flash-es running. If I kill one, the other dies as well, so this is evidently a double-representation of a single running process or application. Nevertheless, it makes me wonder a bit.