Experiments on hybrid superconducting normal-metal structures have revealed that even in the absence of tunnel junctions the onset of superconductivity can lead to a decrease in the electrical conductance by an amount many orders of magnitude greater than e2/h. In this paper, we provide a theory of this phenomenon which shows that it originates from an instability in the four-probe conductance which is absent from two-probe measurements. We compare the zero-bias, zero-temperature four-probe conductances GN and GS of a normal diffusive metal in contact with a superconductor in both the normal (N) and superconducting (S) states, respectively. In the absence of tunnel barriers, the ensemble average of the difference δG = GS-GN vanishes, in agreement with quasiclassical theory. However, we also predict that there exist macroscopic sample specific fluctuations in δG, which lie beyond quasiclassical theory and allow large negative values of δG to occur.