Recent radio pulsar observations have shown that a number of pulsars display interesting long-term periodicities in their spin-down rates. At least some of these pulsars also undergo sharp changes in pulse profile. This has been convincingly attributed to the stars abruptly switching between two different magnetospheric states. The sharpness of these transitions has been taken as evidence against free precession as the mechanism behind the long-term variations. We argue that such a conclusion is premature. By performing a simple best-fitting analysis to the data, we show that the relationship between the observed spin and modulation periods is of approximately the correct form to be accounted for by the free precession of a population of neutron stars with strained crusts, the level of strain being similar in all of the stars, and consistent with the star retaining a memory of a former faster rotation rate. We also provide an argument as to why abrupt magnetospheric changes can occur in precessing stars, and how such changes would serve to magnify the effect of precession in the timing data, making the observation of the precession more likely in those stars where such switching occurs. We describe how future observations could further test the precession hypothesis advanced here.