We perform a set of experiments to study the nonlinear nature of an instability that arises in low-Reynolds-number flow past polymer gels. A layer of a viscous liquid is placed on a polydimethylsiloxane (PDMS) gel in a parallel-plate rheometer which is operated in stress-controlled mode. As the shear stress on the top plate increases, the apparent viscosity stays relatively constant until a transition stress where it sharply increases. If the stress is held at a level slightly above the transition stress, the apparent viscosity oscillates with time. If the stress is increased to a value above the transition stress and then decreased back to zero, the apparent viscosity shows hysteretic behavior. If the stress is instead decreased to a constant value and held there, the apparent viscosity is different from its pretransition value and exhibits sustained oscillations. This can happen even if the stress is held at values below the transition stress. Our observations suggest that the instability studied here is subcritical and leads to a flow that is oscillatory and far from viscometric. The phenomena reported here may be useful in applications such as microfluidics, membrane separations, and polymer processing. They may also provide insight into the rheological behavior of complex fluids that undergo flow-induced gelation.