How do we deal with unlikely witness testimonies? Whether in legal or everyday reasoning, corroborative evidence is generally considered a strong marker of support for the reported hypothesis. However, questions remain regarding how the prior probability, or base rate, of that hypothesis interacts with corroboration. Using a Bayesian network model, we illustrate an "inverse" relationship between the base rate of a hypothesis, and the support provided by corroboration. More precisely, as the base rate of hypothesis becomes more unlikely (and thus there is lower expectation of corroborating testimony), each piece of confirming testimony provides a nonlinear increase in support, relative to a more commonplace hypothesis--assuming independence between witnesses. We show across 3 experiments that lay reasoners consistently fail to account for this impact of (rare) base rates in both diagnostic and intercausal reasoning, resulting in substantial "underestimation" in belief updating. We consider this a novel demonstration of an inverted form of base rate neglect. We highlight the implications of this work for any scenario in which one cannot assume the confirmation or disconfirmation of a reported hypothesis is uniform.