I have an update to this situation. This problem started when the client's account (both files and database) were moved to a private server. I suspect that in that move, the Apapche file owner was changed to something other than the account name.
Normally this probably isn't an issue but with WordPress, the core update and plug-in update functions need to be able to write to the file system and if they are using a file owner other than the one Apache is using, WordPress continually prompts you to supply the information yourself. It's an irritation to say the least.
You can read more about this situation here: http://www.chrisabernethy.com/why-wordpress-asks-connection-info/
From that article: "Many hosting companies will run your apache instance using your user account, and all of your files will be owned by the same account. In those cases, you will probably not have the issue described here."
So I'm guessing that when this client's account was set up for the private server, his Apache instance got a new file owner. This actually broke the site because now the permissions were screwed up and someone had to set the permissions to a less secure state just to get a web browser to read them.
Hopefully this added information helps. My question is how can I get the Apache file owner set BACK to the account name so WordPress is again happy with the permissions it was designed to use?