We consider the consequences for oscillating scalar cold dark matter models of a scalar field phi which oscillates in a potential which deviates slightly from a phi2 potential. We consider a correction to the phi2 potential of the form Kphi2ln(phi2) + lambda(phi)phi4. Upper bounds on \K\ and \lambda)phi)\ are obtained as a function of the phi mass by requiring that the effects of the pressure introduced by the correction to the potential do not disrupt the conventional cold dark matter scenario for structure formation. The results are applied to some cosmological models of recent interest. In particular, it is shown that a scalar identified with the pseudo Goldstone boson of a U(1) symmetry explicitly broken by neutrino masses (suggested iri connection with late-time phase transitions) is inconsistent with the MSW solution of the solar neutrino problem if it is to account for dark matter. In addition it is shown that a cold dark matter model in which the Compton wavelength of a very light scalar particle accounts for the large-scale structure of the Universe is also ruled out by the effects of pressure on structure formation.