One famous example is the Apple team getting a demo of the Graphical User Interface at Xerox PARC. The Apple engineers mistakenly believed that the Xerox interface had a bunch of things it did not, in particular being able to drag overlapping windows on top of each other. The Xerox engineers were supposedly amazed when they saw the results. This isn't quite the same thing - Xerox wasn't staging a fake demo, the Apple guys just mis-remembered or mis-understood what they were looking at. But they still put an insane amount of effort into getting this working, much to everybody's surprise, because they thought that someone else had already done it, so it must have been possible.
Source: Walter Isaacson's Steve Jobs biography. You can probably find this story somewhere on the folklore website.
It's not overlapping windows, but a similar concept. From folklore.org (awesome site, by the way; it made me a fan of that Apple):
Smalltalk didn't even have self-repairing windows - you had to click in them to get
them to repaint, and programs couldn't draw into partially obscured windows.
Bill Atkinson did not know this, so he invented regions as the basis of QuickDraw
and the Window Manager so that he could quickly draw in covered windows and
repaint portions of windows brought to the front.
Ah, I love folklore.org. I devoured every article on the site in about a week. Really makes you wonder what Apple would be like today if Burrell and Andy had had more power to override Jobs (see the "Diagnostic Port" story).
It wasn't overlapping windows, it was the ability to repaint a partially-obscured window without user interaction, plus some confusion because the Xerox Star did not have overlapping windows.
Source: Walter Isaacson's Steve Jobs biography. You can probably find this story somewhere on the folklore website.