Periodically, when I FilletEdge, the inside corners won’t fillet. For example, see the following image: 
Does anyone know why this is?
I get the same, but only if the surfaces are split at the corner (for the screenshot I extracted that surface, but I ran the FilletEdge command while it was one single joined solid).
The split surface corner fails, but the other corner to the right is OK:
From your image it looks like this is also your case?
// Rolf
yes Rolf, this is exactly my problem
Try run MergeFace or MergeAllFaces on the object before running the FilletEdge, then you may have a better outcome with the FilletEdge command.
// Rolf
This obviously does only work with planar surfaces…
For other cases you will need to manually fix fillededges trimming shortcomings.
Norbert
True. Hence my wording “may” 
// Rolf