The structure and lattice dynamics of rock-salt thermoelectric materials SnTe and PbTe are investigated with single crystal and powder neutron diffraction, inelastic neutron scattering (INS), and first-principles simulations. Our first-principles calculations of the radial distribution function (RDF) in both SnTe and PbTe show a clear asymmetry in the first nearest-neighbor (1NN) peak, which increases with temperature, in agreement with recent experimental reports. We show that this peak asymmetry for the 1NN Sn-Te or Pb-Te bond results from large-amplitude anharmonic vibrations (phonons). No atomic off-centering is found in our simulations. In addition, the atomic mean square displacements derived from our diffraction data reveal stiffer bonding at the anion site, in good agreement with the partial phonon densities of states from INS, and first-principles calculations. These results provide clear evidence for large-amplitude anharmonic phonons associated with the resonant bonding leading to the ferroelectric instability.