Many dilute surfactant solutions containing long, thin, wormlike micelles undergo a shear-thickening transition at which the apparent viscosity jumps to a much higher value due to the formation of shear-induced structures. However, the behavior of the normal-stress differences N-1 and N-2 has not been studied at the low-shear rates and shear stresses ( < 0.1 Pa) at which shear thickening occurs in most of the commonly studied systems, such as aqueous solutions of cetyltrimethylammonium bromide (CTAB) and sodium salicylate (NaSS). For dilute equimolar solutions of CTAB/NaSS, the validity of measurements of the shear flow properties eta, N-1 and N-2 via cone-and-plate rheometry is assessed by systematically varying the cone angle, and by measuring the pressure profile during cone-and-plate flows. The dependence on the cone angle of the apparent viscosity value is consistent with a stratified fluid structure, as shown to occur in Couette rheometers by Hu and co-workers [Hu, Y. T. et al., J. Rheol. 42, 1209-1226 (1998)]. However, N-1 and N-2 values show no obvious dependence on cone angle, and the occurrence of flow stratification is surprisingly difficult to detect from the pressure profile. As the shear rate is increased, the measured pressure profiles are suggestive of a gradual transition. from viscometric to nonviscometric flow. By combining the results presented here with earlier results by Huang and co-workers [Huang, C.-M. et al., in Proceedings of the XII International Conference on Rheology, edited by A. Ait-Kadi et al. (Canadian Rheology Group, Quebec City, 1996)], we conclude that: the same wormy micelle solution exhibits at least two flow transitions: that might be described as "shear thickening." In the shear-rate range of viscometric cone-and-plate flows, the measured normal-stress ratio -N-2/N-1 for dilute solutions of CTAB/NaSS wormy micelles is similar to the value observed for moderately entangled solutions of synthetic polymers.