That did it!!! Many thanks.

Maybe it just gets skinned over. I'm pretty new to SketchUp, so I have not sdjusted any settings from the default settings on Pro8. An outline of the original inset box still shows up when I do the round edges, but it appears to be back in the same plane as the outer box.
If you are referring to the Round Edge settings, I have been using 3/16" radius edges with 6 segments.

If you meant the settings on RoundEdge,I was using an edge radius of 3/16" and have tried a number of different segments for rounding, but 6 segments would be fine.

I am having trouble with putting round edges on an inset box, within a box. When I try to round the edges, the inset box disappears. I have attached a very simple model which will illustrate my problem if you select the edges and round them. Any idea what I need to do differently in order to make all the edges (both inside and outside edges) become rounded?

Inset Box file