The high-pressure melting of Li and Na has been studied using ab initio calculations of the lattice dynamics. It has been shown that the recently discovered anomalous melting of Na is adequately explained by the phonon spectrum behavior and, accordingly, the thermal vibration amplitudes under compression. In a simple approach using the Lindemann criterion, the nonmonotonic behavior of the melting curve Tm(p) of Na has been quantitatively described within very wide pressure and temperature ranges, and, in particular, the melting temperature drop at p ∼ 1 Mbar down to values lower than those at normal pressure. This approach leads to a nonphysical discontinuity of the melting curve Tm(p) of Li near the bcc-fcc-liquid triple point. This is due to the "softness" of the phonon spectrum of the bcc phase of Li that is the necessary condition for the existence of the high-temperature bcc phase. The melting of Na and Li is used as an example to determine why the Lindemann criterion is efficient in some situations and is inapplicable in the other cases.