We attempt to resolve the controversy concerning the minimum temperature of Ly alpha clouds. The results by Pettini et al. are scrutinized, namely that Ly alpha absorption lines have Doppler parameters with a median of about 17 km s-1, values below 10 km s-1, and that there is a strong intrinsic correlation between Doppler parameter and column density. The AAT/UCLES spectra of the z(em) = 2.56 QSO 2206-199 have been reexamined. A search for heavy-element lines at redshifts corresponding to known absorption systems and for previously unidentified heavy-element systems results in possible new identifications for some of the narrowest lines in Pettini et al.'s Ly alpha sample, but few of the narrower lines could be attributed to heavy elements. For the remaining Ly alpha lines, an apparent strong correlation between Doppler parameter b and H I column density N is found, along with the presence of a substantial number of narrow Ly alpha lines. However, using numerical simulations of 2206-199 we show that both the presence of very narrow Ly alpha lines and the apparent correlation between b and N(H I) for the weakest lines may be artefacts of the line selection and fitting procedure and could arise if in the underlying distribution the mean Doppler parameter is 30 km s-1 and there is no intrinsic correlation between Doppler parameter and H I column density. The contradictory findings of Pettini et al. and Carswell et al. may be explained in terms of higher signal-to-noise ratio in the latter study, in the sense that the Carswell et al. data are less sensitive to the noise-induced distortion of the underlying parameter distribution for the range of column densities measured. We thus find that the 2206-199 data are consistent with previous high-resolution analyses of QSO spectra, with little evidence for a significant population of cool clouds with Doppler parameters less than about 15 km s-1.