This paper presents a general analysis and a concrete example of the catastrophic case of a discontinuity-induced bifurcation in so-called Filippov nonsmooth dynamical systems. Such systems are characterized by discontinuous jumps in the right-hand sides of differential equations across a phase space boundary and are often used as physical models of stick-slip motion and relay control. Sliding bifurcations of periodic orbits have recently been shown to underlie the onset of complex dynamics including chaos. In contrast to previously analyzed cases, in this work a periodic orbit is assumed to graze the boundary of a repelling sliding region, resulting in its abrupt destruction without any precursive change in its stability or period. Necessary conditions for the occurrence of such catastrophic grazing-sliding bifurcations are derived. The analysis is illustrated in a piecewise-smooth model of a stripline resonator, where it can account for the abrupt onset of self-modulating current fluctuations. The resonator device is based around a ring of NbN containing a microbridge bottleneck, whose switching between normal and super conducting states can be modeled as discontinuous, and whose fast temperature versus slow current fluctuations are modeled by a slow-fast timescale separation in the dynamics. By approximating the slow component as Filippov sliding, explicit conditions are derived for catastrophic grazing-sliding bifurcations, which can be traced out as parameters vary. The results are shown to agree well with simulations of the slow-fast model and to offer a simple explanation of one of the key features of this experimental device.