From a sample of 84 local barred, moderately inclined disc galaxies, we determine the fraction that hosts boxy or peanut-shaped (B/P) bulges (the vertically thickened inner parts of bars). We find that the frequency of B/P bulges in barred galaxies is a very strong function of stellar mass: 79 per cent of the bars in galaxies with log (M*/M_{sun}_)w>=10.4 have B/P bulges, while only 12 per cent of those in lower mass galaxies do. (We find a similar dependence in data published by Yoshino & Yamauchi for edge-on galaxies.) There are also strong trends with other galaxy parameters - e.g. Hubble type: 77 per cent of S0-Sbc bars, but only 15 per cent of Sc-Sd bars, have B/P bulges - but these appear to be side effects of the correlations of these parameters with stellar mass. In particular, despite indications from models that a high gas content can suppress bar buckling, we find no evidence that the (atomic) gas mass ratio M_HI+He_/M* affects the presence of B/P bulges, once the stellar-mass dependence is controlled for. The semimajor axes of B/P bulges range from one-quarter to three-quarters of the full bar size, with a mean of R_box_/L_bar_=0.42+/-0.09 and R_box_/a_{epsilon}_=0.53+/-0.12 (where R_box_ is the size of the B/P bulge and a_{epsilon}_ and L_bar_ are lower and upper limits on the size of the bar).