ABSTRACT We show that magnetar models for ultraluminous X-ray sources (ULXs) have serious internal inconsistencies. The magnetic fields required to increase the limiting luminosity for radiation pressure above the observed (assumed isotropic) luminosities are completely incompatible with the spin-up rates observed for pulsing ULXs. We note that at least one normal Be-star + neutron star system, with a standard (non-magnetar) field, is observed to become a ULX during a large outburst and return to its previous Be-star binary state afterwards. We note further that recent polarimetric observations of the well-studied binary Cyg X-3 reveal that it produces strong emission directed away from the observer, in line with theoretical predictions of its total accretion luminosity from evolutionary arguments. We conclude that the most likely explanation for ULX behaviour involves radiation beaming by accretion disc winds. A large fraction of X-ray binaries must pass through a ULX state in the course of their evolution.