Comments on the Wordpress support forum pointed to perhaps the server being overwhelmed and reverting to a "safe" download behaviour.
Either way, I think it's time to look at the options. I'm currently looking at DreamHost. Their services seem really good (use a wiki for support info, have SSH access and a bunch of other nicenesses that GoDaddy doesn't). Are there better options? Should I be concerned about Patriot Act and DMCA implications with an American hosting provider?
I'm also wondering about what the migration process is. GoDaddy hosts the DNS as well, so how smooth is it to move that to another provider?