Then the cache got updated as well. The best way to go is as 40hz suggested: changing the DNS servers your connection to the internet uses.
I looked at an article that Shades gave me on this stuff. I might have done that if I were having terrible overall problems. But it's mostly one single site, with a few occasional flukes elsewhere, and the downside of messing this up bothers me.
So today I went a different route. I went to a proxy server at proxybrowsing.com and pasted the url in there, and it seems to work much better (at least for now!)
My logic is that whatever fluky local glitch in my exact connection, it doesn't exist from their side. So we'll see!