We investigate the flow resistance of a droplet trapped at a constriction in a microcavity located at a microchannel bifurcation as a function of system parameters including capillary number, drop confinement, and viscosity ratio. Using a combination of experiments and volume-of-fluid numerical simulations, we measure the hydrodynamic resistance of the trapped drop and connect it to drop deformation in the microcavity. For drop sizes smaller than the microcavity, we observe a bistable behavior in terms of the resistance of the trapped drop as a function of capillary number. For these underfilled drops, we find that the resistance is low at small capillary number (Ca < 10−3) and jumps to high resistance at a threshold capillary number. For drops equal to the microcavity size, we observe that the bistability vanishes and the drop resistance is of similar magnitude as that of underfilled drops at large capillary number. To explain these findings, we use confocal microscopy and simulations to obtain three-dimensional views of the drop deformation and continuous phase fluid in the microcavity. We observe that the low resistance is due to negligible drop deformation and unobstructed flow of continuous phase through the constriction. The high resistance is due to the drop interface protruding into the constriction restricting the flow of continuous phase through the gutters. Taken together, our results indicate that a trapped drop at a bifurcation can act as a nonlinear resistor and could be potentially used as a soft switch to control droplet trajectories in microfluidic devices.