Converse of ML inequality for contour integrals

Question

If the ML inequality estimates a region of an integral in the complex plane to be zero then that's the actual value of the integral, and I've been using this for evaluating some integrals along the real line using contour integration. But is the converse true, that if that part of an integral approaches 0 then the ML inequality will say so. Does this estimation ever fail, and if so are there other methods that do work more often?