Computations are reported for a shear thinning, elastic fluid in axisymmetric stagnation flows, in an attempt to predict flow birefringence data on a 5 wt % high molecular weight polystyrene solution. For multimode Phan-Thien-Tanner (PTT) and Giesekus models, several parameter sets are chosen in which the degree of elongational hardening is systematically adjusted. The primary objective is to see whether the extensional properties of the fluid can be inferred from comparisons between these computations with normal stress measurements in flows dominated by uniaxial extension. Using the PTT model it is possible td vary extensional predictions while maintaining a good fit of shear viscosity data. Up to moderate Weissenberg number, these comparisons suggest a modest degree of extension hardening. Computations in rear stagnation flow using the PTT model fail at the intermediate Weissenberg number. A multimode Giesekus model with parameters determined only by fitting shear data performs exceptionally well in forward stagnation flow at all Weissenberg numbers, but is deficient in rear stagnation flow at the two highest experimental Weissenberg numbers. Attempts to selectively enhance extensional hardening at high rates do improve normal stress predictions in the rear stagnation region, but only at the expense of degraded predictions of shear stress in rear stagnation flow, and all stresses in forward stagnation flow.