ABSTRACT In this paper, we use the Herschel Reference Survey to make a direct test of the hypothesis that the growth of a stellar bulge leads to a reduction in the star formation efficiency of a galaxy (or conversely a growth in the gas-depletion time-scale) as a result of the stabilization of the gaseous disc by the gravitational field of the bulge. We find a strong correlation between star formation efficiency and specific star formation rate in galaxies without prominent bulges and in galaxies of the same morphological type, showing that there must be some other process besides the growth of a bulge that reduces the star formation efficiency in galaxies. However, we also find that galaxies with more prominent bulges (Hubble types E to Sab) do have significantly lower star formation efficiencies than galaxies with later morphological types, which is at least consistent with the hypothesis that the growth of a bulge leads to the reduction in the star formation efficiency. The answer to the question in the title is therefore yes and no: bulges may reduce the star formation efficiency in galaxies but there must also be some other process at work. We also find that there is a significant but small difference in the star formation efficiencies of galaxies with and without bars, in the sense that galaxies with bars have slightly higher star formation efficiencies.