We have demonstrated that the observed elastic anomalies and lattice instabilities in A15 materials are well described using a standard LAPW/LDA electronic structure approach. We have shown that A15 structure V3Si adn Nb3Sn are unstable (the total energy exhibiting a double-well structure) with regard to shear instabilities with elastic moduli C11-C12 and C44 at T=0 (and that V3Ge is weakly stable). It is crucial to allow sublattice distortions in these calculations. Several optic phonon modes corresponding to sublattice distortions that are consistent with the known low-temperature tetragonal phase are unstable as well. At high temperature the well structure is washed out and the atoms assume the A15 structure, and we obtain excellent agreement with experiment for the high-temperature elastic constants.