I'm afraid I don't know the origin of this way of phrasing and I've not seen it before, however I was surprised to see that the answers here consider 'fails if' equivalent to 'passes if', as I quite strongly consider the two to be very different from each-other and also my opinion is that 'fails if' is slightly superior to 'passes if' for manual tests anyway - if I think about an automated test it seems only able to assert a 'passes if' or a 'don't fail if'.
Taking the example from glowcoder's answer, a requirement for a search box that is prompted to the user. In a test step that tests this condition, we could say 'the test fails if the search box is not prompted to the user' or 'the test passes if the search box is prompted to the user'. They seem equivalent - both do not fail if the search box is prompted to the user but consider the following results:
Actual Result: The search box is prompted to the user but there's an error on the page
- 'fails if' pass or fail: undetermined
- 'passes if' pass or fail: pass
Actual Result: The search box is prompted to the user but the label for Search is spelt 'Saerch'
- 'fails if' pass or fail: undetermined
- 'passes if' pass or fail: pass
Actual Result: The search box is prompted to the user but it takes more than ten seconds for the search box to load
- 'fails if' pass or fail: undetermined
- 'passes if' pass or fail: pass
Hopefully there's not much in it, as any manual test will hopefully be run by someone with the ability to identify that although the pass criteria are met, if errors occur when the search box loads, then the step cannot really be considered to be passing. It just feels to me that 'passes if' may have an increased tendency to encourage confirmation bias in the tester, in that they only need to look for one pass condition to be able to move onto the next step with confidence.
Personally, I don't use either terminology in my test scripts tending as I do to use a more dispassionate and less leading terminology like 'There will be a search box that is prompted to the user'. I have to able to safely assume that the testers I work with are able to look around the software under test and read far more than I could ever write into a test step. In my opinion there are always many more possible failure conditions than there are easily identifiable passing ones.