The output vs input behavior of the NCP500 is perfectly documented in its datasheet. That's why I mentioned it.
I didn't mean to suggest otherwise. The NCP500 is unusual for mentioning the input vs output characteristic on the data sheet.
Granted it's one of the few that document this as well. Actually a lot of the LDOs in the NCPxxx line are similar in that respect. I've used them on occasion and they work according to the specs. The nice thing is that the drop-out voltage when Vin is below Vreg is very low, so for battery operated stuff, that's nice.
Of course, there are other situations in which this behavior is, on the opposite, not wanted. When your design is not guaranteed to work below a specified voltage, you'll want the regulator (or another supervisory circuit) to shutdown your circuit instead of powering it at unsafe voltages.