The Bagnold hypothesis has been a major tool in the development of mechanistic models of bed load transport. According to the hypothesis, a necessary constraint for the maintenance of equilibrium bed load transport is that the fluid shear stress at the bed must be reduced to the critical, or threshold value associated with incipient motion of grains. The constraint determines a functional form for the areal concentration of bed load grains as a function of flow parameters. It is shown here, however, that the Bagnold hypothesis breaks down when applied to equilibrium bed load transport on beds with transverse slopes above a relatively modest value that is well below the angle of repose. In particular, under such conditions it is shown that no areal concentration, no matter how large, is sufficient to reduce the fluid shear stress at the bed to the critical value. This failure motivates the abandonment of the Bagnold constraint, even for nearly horizontal beds. The framework presented here, however, provides a natural basis for an entrainment-based model of sediment transport that neither satisfies nor suffers from the drawbacks of the Bagnold constraint.