Localization of deformation and failure, a complete loss of stress carrying capacity, is studied for two rate dependent constitutive relations: (i) a KelvinVoigt solid and (ii) a viscoplastic solid. A planar block infinite in one direction is subjected to monotonically increasing shear displacements at a fixed rate. Geometry changes are neglected and attention is confined to quasi-static loading conditions. For the KelvinVoigt solid, localization precedes failure if there is hardening outside the band and softening inside the band while failure precedes localization if there is softening both inside and outside the band. For the viscoplastic solid, localization precedes failure when there is softening inside the band regardless of the sign of the hardening parameter outside band. For the KelvinVoigt solid, it is found that the localization time (or strain) varies logarithmically with the band thickness for small values of band thickness while the time (or strain) to a complete loss of stress carrying capacity has, in general, a different scaling with band thickness. For the viscoplastic solid, with plastic dissipation outside the band as well as inside the band, the strain and the total plastic dissipation to failure are nearly independent of band thickness for sufficiently small thickness values, with what is sufficiently small decreasing with decreasing rate sensitivity. Possible implications for grid based modeling of localization and failure are discussed.