In the stetcher editor, on a cabinet that I had downloaded from one of the libaries, I couldn't delete one of the drawer stretchers. I could highlight it, but when I tried to delete it, it stayed. With it highlighted, I'd pick 'adjust stretcher', and the image, and adjustments would come up for the rear nailer instead. Not clear on how to get rid of it.
Since I have no idea what this cabinet looks like I will guess that you are actually trying to delete a shelf or a partition, so you need to select that editor.
If you hover over any object, such as a shelf or a stretcher the software will tell you what kind of part it is along with all of the other information you need to know. Your "stretcher" was actually a fixed shelf with a back inset. The parts need to be operated on in their related sub-editor.