Have finally reached a solution or success with this problem, possibly with a little input from someone on a GIS listerv that I am on.
One thing to remember, always refresh in ArcCatalog, although that wasn't the source of my problem.
Will cut and paste a suggestion and my later actions in case somebody comes across a similar problem.
"There are a couple of things I have found when converting CAD to GIS. First, try turning off background processing under Geoprocessing options. It has been my experience that the CAD Conversion tools do not like background processing."
I did this, and I still had the same problem for several attempts afterwards, but kept this dis-enabled, and finally met with success. So who knows?
One less efficient method: I went directly to the .dwg file and did export to a geodatabase, and that actually worked, but did not create a feature dataset, just several feature classes, which I then had to re-import into a feature dataset.
After I did that, I re-tried the CAD to Geodatabase tool, and it finally worked. And worked again...... One factor could have been a complete shutdown overnight, but not sure about that as the real solution. Only the GIS gods know.