We study the time-dependent compressible flow of a Newtonian fluid in slits using an arbitrary nonlinear slip law relating the shear stress to the velocity at the wall. This slip law exhibits a maximum and a minimum and so does the flow curve. According to one-dimensional stability analyses, the steady-state solutions are unstable if the slope of the flow curve is negative. The two-dimensional flow problem is solved using finite elements for the space discretization and a standard fully implicit scheme for the time discretization. When compressibility is taken into account and the volumetric flow rate at the inlet is in the unstable regime, we obtain self-sustained oscillations of the pressure drop and of the mass flow rate at the exit, similar to those observed with the stick-slip instability. The effects of compressibility and of the length of the slit on the amplitude and the frequency of the oscillations are also examined.