We present a new test of the widths of the local line profiles from the accretion discs of cataclysmic variable stars. Our method is based upon the near coincidence of the Ca II H line at 3968.5 A and the HE line at 3970.1 A. If at any point in the disc the profiles of these lines overlap, the total flux seen will in general not be the sum of the individual fluxes; if the profiles do not overlap, the fluxes will add linearly. This diagnostic applies independently of the Doppler broadening from the disc. We estimate the 'true' HE flux from interpolation between neighbouring members of the Balmer series. Subtracting this flux from the combined flux of the HE/Ca II H blend leaves the Ca II H flux which can be compared with the flux of the isolated Ca II K line. We find that the Ca II infrared triplet lines at 8498, 8542 and 8662 A show optically thick flux ratios close to 1, implying that the H/K ratio should also be close to 1, whereas the observed ratios are close to 0.5. Our results show that although HE is separated by 120 km S-1 from Ca II H, compared with a sound speed of only ~1O km s-I, the lines are overlapping so that the Ca II H emission is suppressed. Thus some form of broadening beyond the purely thermal is required. The extra broadening agrees with evidence for supersonic turbulence from ultraviolet observations of the dwarf nova OY Car. However, any other form of broadening of sufficient strength will suffice. We consider Stark broadening and show that it can be significant if the lines form at baryon densities above '" 5 x 1013 cm-3 and temperatures above '" 8000 K; it is not possible to say at present if such conditions are likely. The supersonic line broadening was present in all the quiescent dwarf novae we were able to observe (other than U Gem, which we could not measure because of severe blending). The broad local line profile implied by our data (FWHM > 240 km S-1) may explain the absence of double-peaked profiles in many dwarf novae.