Possible issue where .NET Framework 4 setup reports success but fails to update mscoree.dll behind the scenes

I have heard from a few customers since the release of the .NET Framework 4 who have installed the .NET Framework 4 on Windows Vista or higher and setup reported success, but they see an error when running the .NET Framework setup verification tool.  I wanted to describe this scenario in a bit more detail, including how to diagnose whether or not your computer is running into this issue based on