Polarized Raman scattering on single crystals of NaAlH4 has been used to determine the symmetry properties and frequencies of the Raman-active vibrational modes over the temperature range from 300 to 425 K, i.e., up to the melting point T-melt. Significant softening (by up to 6%) is observed in the modes involving rigid translations of Na+ cations and translations and librations of AlH4-. Surprisingly, the data indicate mode softening of less than 1.5% for the Al-H stretching and Al-H bending modes of the AlH4- anion. These results show that the AlH4- anion remains a stable structural entity even near the melting point. First-principles linear response calculations of phonon mode frequencies are in reasonably good agreement with the Raman results. The phonon mode Gruneisen parameters, calculated using the quasiharmonic approximation, are found to be significantly higher for the translational and librational modes than for the Al-H bending and stretching modes, but cannot account quantitatively for the dramatic softening observed near T-melt in the former two types of modes, suggesting an essentially anharmonic mechanism. The effect of zero-point vibrations on the calculated lattice parameters is found to be large (expansion by 1.2 and 1.5 % in the a and c parameters, respectively), as expected for a compound with many light elements. We discuss the implications of the observed mode softening for the kinetics of hydrogen release and hypothesize that breaking up the AlH4- anions is the rate limiting step. The enhanced kinetics of absorption and desorption in Ti-doped NaAlH4 powders is attributed to the effectiveness of Ti in promoting the breakup of the AlH4- anions.