Oh, this is a lovely catch all, isn't it? It did help me discover a couple things.
1. The MAC address in our Autodesk Inventor 2010 license file was incorrectly entered when the license was generated.
2. This particular error means anything and everything is wrong.
Here's what went wrong on our installation. For some reason, when the licensing directory at C:\Program Files\Autodesk Network License Manager\ was created, the executables LMTOOLS were added, but not the additional directories. So there's no sub-directory for license files and no debug.log file.
The last one was our problem. I incorrectly assumed that creating the path under Config Services would create the file. It didn't. So every time I tried to start my licensing server I got the dreaded "lmgrd is not running: Cannot connect to license server" error message. As a work around I created a file called debug.log, recreated the path to the debug log file and the server works like a charm.
This may also apply to ESRI applications and other programs using FlexLM and LMTOOLS.