This paper introduces a novel bootstrap procedure to perform inference in a wide class of partially identified econometric models. We consider econometric models defined by finitely many weak moment inequalities, which encompass many applications of economic interest. The objective of our inferential procedure is to cover the identified set with a prespecified probability. We compare our bootstrap procedure, a competing asymptotic approximation, and subsampling procedures in terms of the rate at which they achieve the desired coverage level, also known as the error in the coverage probability. Under certain conditions, we show that our bootstrap procedure and the asymptotic approximation have the same order of error in the coverage probability, which is smaller than that obtained by using subsampling. This implies that inference based on our bootstrap and asymptotic approximation should eventually be more precise than inference based on subsampling. A Monte Carlo study confirms this finding in a small sample simulation.