The mechanical response of yield-stress materials below the yield point remains a subject of debate. Two of the most widely used constitutive models for these materials offer fundamentally conflicting views: one permits plastic flow at all stress levels, while the other assumes entirely recoverable viscoelasticity below yield. Using parallel superposition rheometry, we test the subyield behavior of a microgel and an emulsion. When residual slip effects are properly accounted for, both fluids exhibit bounded, periodic strain responses, offering compelling evidence that they do not flow in the studied regime. Our results indicate that the subyield regime is underpinned by nonlinear viscoelasticity and underscore the need for improved constitutive relations that capture such effects without treating yielding as a precursor for nonlinearity.