We study and test a class of boundedly rational models of decision making which rely on sequential eliminative heuristics. We formalize two sequential decision procedures, both inspired by plausible models popular among several psychologists and marketing scientists. However we follow a standard `revealed preference' economic approach by fully characterizing these procedures by few, simple and testable conditions on observed choice. Then we test the models (as well as the standard utility maximization model) with experimental data. We find that the large majority of individuals behave in a way consistent with one of our procedures, and inconsistent with the utility maximization model.