After centuries of striving for structural rigidity, engineers and scientists alike are increasingly looking to harness the deformation, buckling, and failure of soft materials for functionality. In fluidic devices, soft deformable components that respond to the flow have the advantage of being passive; they do not require external actuation. Harnessing flow-induced deformation for passive functionality provides a means of developing flow analogs of electronic circuit components such as fluidic diodes and capacitors. The electronic component that has so far been overlooked in the microfluidics literature—the fuse—is a passive safety device that relies on a controlled failure mechanism (melting) to protect a circuit from overcurrent. Here, we describe how a compliant Hele-Shaw cell behaves in a manner analogous to the electrical fuse; above a critical flux, the flow-induced deformation of the cell blocks the outflow, interrupting (choking) the flow. In particular, the pressure distribution within the fluid applies a spatially variant normal force to the soft boundary, which causes nonuniform deformation. As a consequence of lateral confinement and incompressibility of the soft material, this flow-induced elastic deformation manifests as bulging near the cell outflow; bulges that come into contact with the rigid cell roof interrupt the flow. We identify two nondimensional parameters that govern the central deflection and the choking of the cell, respectively. This study therefore provides the mechanical foundations for engineering passive-flow limiters into fluidic devices.