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.
I've heard that during Noam Shazeer's Google interview, he was asked how to implement a spelling corrector. He described a scheme for statistically verifying queries against logs of what other users have typed. The interviewer (Paul Buchheit IIRC) quickly realized that his solution was better than what Google was using at the time. Noam was hired and asked to implement his interview question in production, and the resulting algorithms form the basis of the spell corrector that's used on Google today (though I hear the code was completely rewritten a few years ago).
Always better to ask unsolved problems with than solved problems in an interview. I wish Google still interviewed this way, instead of toy textbook homework problems.
Based on a press-release for the capabilities of a new graphics card that would have blown his project out of the water, an engineer spent a long time thinking about what they must be doing to get that sort of performance. He figured they were using a FIFO somewhere in the circuitry, and had enough space to implement a 1-deep FIFO for a certain operation, figuring that it wasn't enough. Turns out, his improved chip beat the benchmarks that were in the original press release because they had implemented a FIFO for a different operation that had less impact on performance.
Not directly related but there is the story of George Dantzig, a graduate student at Berkeley.
Arriving late to a class he quickly jotted down two equations left on the board, thinking they were homework. Several days later he hands the solutions to his professor. When his professor finally marks the problems he realises that George just proved two statistical theorems which had been unsolved up to that point.
There was a recent screenshot of Uncharted (a modern PS3 game franchise) highlighting the differences between the E3 build and the final shipping build.
