In case anyone runs into the same problem: it appears that the ASP.NET MVC 3 installer depends on being able to install a particular version of NuGet.
If you have a newer version installed, that part of the installer will fail, leading the rest of the install to also fail.
The solution is to uninstall NuGet, install ASP.NET MVC 3, and then update NuGet to the version you were on.
