Reversible shear thickening is measured in model colloidal suspensions of silica that exhibit near hard‐sphere behavior. Flow dichroism measurements and the colloidal stress‐optical relationship prove that hydrodynamic interactions drive the shear thickening transition. Turbidity and flow‐small angle neutron scattering (flow‐SANS) demonstrate that particles cluster reversibly in the shear thickened state. Further, SANS measurements show that shear thickening occurs without any shear‐induced order‐disorder transition, in contrast to observations for dispersions of charged colloids. A simple force balance predicts the scaling of the critical stress for the onset of shear thickening with particle size and volume fraction. Measurements on bimodal mixtures verify the scaling laws derived from the force balance and provide a strategy for controlling the shear thickening transition.