The viscosity of concentrated shear thickening dispersions was measured as a function of shear rate, Couette cylinder size, and time. The level of the low shear rate viscosity, which was found to be independent of system size and time, could be predicted by the equation of Frankel and Acrivos. At shear rates above the critical shear rate for shear thickening in highly concentrated (φ≳0.57) dispersions of monodisperse particles strong viscosity instabilities were detected, together with a dependence on cylinder size. The instabilities are attributed to reversible order–disorder transitions, e.g., from strings to clusters. This dependence on cylinder size is due to wall slip, slipping planes in the dispersion, and even plug flow in the gap. With less concentrated or polydisperse dispersions the effects are much less severe but there is thixotropy, probably due to a reordering of the dispersion.