I don't think this counts, in that the predictions of relativity were not needed in the design. Relativity is needed to explain why certain phenomena crucial to the design occur, but the phenomena themselves were already known at the time of design. Another similar example is that relativity is needed to explain the color of gold. If all the evidence you had for relativity was of this form, and then it turned out relativity was wrong, you wouldn't be too surprised - you'd have gotten your theory wrong, but you wouldn't be saying "How the hell can batteries work, then?!" You still wouldn't know how car batteries worked, of course, but it wouldn't seem impossible that they should work, since they were based on phenomena known prior to relativity. By contrast, learning that relativity was wrong should make GPS seem impossible, since relativity was needed to predict that it would work (in its current form, with the relativistic calculations included) in the first place. (Here of course by "wrong" I mean "significantly off in these situations" rather than just "not literally true".)