We present experimental results on dense corn-starch suspensions as examples of non-Brownian, nearly-hard particles that undergo continuous and discontinuous shear thickening (CST and 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. For further information, see the related pre-print: Hermes et al (2015) "Unsteady flow and particle migration in dense, non-Brownian suspensions" at http://arxiv.org/pdf/1511.08011v1.pdf (N.B. figure numbers in the pre-print are not necessarily the same as those on the filenames in this dataset).