Long story short, it turns out the WebDAV support in Windows 7 doesn't seem to like certain characters in paths, specifically space in this case (although I assume there are others). In this case it didn't like that the Document Library was at:
This is with WebClnt.dll 6.1.7601.21728, which is the most recent I could track down.
The work around is to rename the document library. Since you can't do this directly from the SharePoint web interface, I had to do it using SharePoint Workspaces. Once the library was renamed, it's working properly now.