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We investigate the source of the large variations in the observed detection efficiencies of superconducting nanowire single-photon detectors between many nominally identical devices. Through both electrical and optical measurements, we infer that these variations arise from “constrictions:” highly localized regions of the nanowires where the effective cross-sectional area for superconducting current is reduced. These constrictions limit the bias-current density to well below its critical value over the remainder of the wire, and thus prevent the detection efficiency from reaching the high values that occur in these devices when they are biased near the critical current density.