A criterion to predict instability in rate-dependent materials is developed. It is implemented for PET fibers in tension at three temperatures: 60, 75, and 80 degrees C. At 60 degrees C, necking is always observed, whereas at 80 degrees C, the deformation is uniform, and 75 degrees C marks a transition region, where necking is observed only at higher speeds and the deformation is otherwise uniform. As a necessary tool in the implementation of this criterion, the stress-strain behavior of PET is modeled using a combination of an Eyring process, a Gaussian network, and a linear elastic element. The resulting instability model gives predictions that are generally consistent with the experimental observations at all temperatures.