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 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.