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We present experimental results on dense corn-starch suspensions as examples of non-Brownian, nearly hard particles that undergo continuous and discontinuous shear thickening (DST) at intermediate and high densities, respectively. Our results offer strong support for recent theories involving a stress-dependent effective contact friction among particles. We show, however, that in the DST regime, where theory might lead one to expect steady-state shear bands oriented layerwise along the vorticity axis, the real flow is unsteady. To explain this, we argue that steady-state banding is generically ruled out by the requirement that, for hard non-Brownian particles, the solvent pressure and the normal-normal component of the particle stress must balance separately across the interface between bands. (Otherwise, there is an unbalanced migration flux.) However, long-lived transient shear bands remain possible.