As part of a continuing study of the lithium abundances in pre-main sequence (PMS) low mass stars, we present Li observations for 13 K-type and 3 M-type stars belonging to the Pleiades and alpha Persei open clusters, respectively. We have supplemented our measurements with previous data taken from the literature and derived abundances in LTE and NLTE for 27 stars. We confirm that a Li spread exists which is connected to rotation for stars with approximately 0.7 - 0.9 solar mass: fast rotators show, in general, more Li than slowly rotating stars by about one order of magnitude. High rotational velocity seems to inhibit the PMS Li depletion in these stars. We speculate on several physical hypotheses to explain this effect. A spread is also present among lower mass stars, but it does not seem to be related to rotational velocity. Convective mixing appears to be the dominant process of Li depletion in the lowest mass range considered in this work.