We have combined rheology and small-angle neutron scattering experiments to investigate dilute aqueous solution of cetyltrimethylammonium tosylate (CTAT). Rheology experiments have been performed under controlled strain conditions as function of concentration (phi = 0.05 - 1%) and temperature (T = 23-45 degrees C). At low surfactant concentration and under steady shear conditions, the CTAT exhibits a continuous increase of the apparent viscosity with increasing shear rates, also termed a shear-thickening transition. The critical shear rate for shear thickening varies strongly with the temperature according to an Arrhenius behavior, but very weakly with concentration. On the same solutions that were studied by rheology, neutron scattering was performed at rest to search for a correlation between the rheological features of the shear-thickening transition and the equilibrium micellar structure. We have found that (i) for all concentrations that display the thickening transition, the local morphology is that of cylindrical micelles, with a radius R = 21.0 +/- 0.5 Angstrom; (ii) all spectra exhibit a scattering correlation peak due to strong electrostatic repulsion between charged micelles; (iii) the shear-thickening transition in CTAT displays the same rheological features for solutions prepared either in the dilute (phi < phi*) or in the semidilute (phi* < phi < 0.8%) regimes.