An analysis of cup-anemometer dynamics has been carried out in order to determine whether the mean-wind velocity can have a negative bias. This would be contrary to the general belief that cup anemometers always overspeed. Compared to prior analyses, the effect of a possible nonlinearity of the calibration function is included. The conclusion is that neither longitudinal nor lateral velocity fluctuations can contribute significantly to a negative bias. However, if a cup anemometer has an angular response that falls below the ideal cosine response, there will, as demonstrated in the concluding discussion, be a negative contribution from the vertical velocity fluctuations to the total bias, and this contribution may even outbalance the positive contributions from the longitudinal velocity fluctuations. Concrete evidence of such exotic cup anemometer behaviour has not been reported in the literature.