Thanks for your help folks!
My first call to GoDaddy support got me a guy who claimed that renaming the php5.ini file would fix the problem. The file had only one command in it, the universal "cgi.fix_pathinfo = 1" command. Who can't get along with that?
I read somewhere that the command has to do with helping WP to do permalinks. I deleted the file, and still did not have success.
I called GoDaddy back. My second call got me a guy who claimed having Fast CGI enabled (when it was not being used, he claimed) was causing a problem. I am not knowledgeable about CGI or Fast CGI, and I granted him permission to disable it. He said I have power in their online panel to enable or disable it myself, but I don't know how to do that.
As soon as he disabled it, the problem seemed to go away.
Then my WP sites that were stored in subfolders on the hosting server seem to be able to do permalinks. However, I then upgraded an older WP blog (which was stored in the root folder of the hosting server) to 3.0. I tried to access pages that happen to require permalinks, and they would not function. I get 404 errors.
I tried putting the php5.ini file back, but that seems to re-cause the serious problems I was trying to get fixed. I am now moving the root folder WP blog into a subfolder to see if I can get it to work. Argh.