By performing molecular dynamics simulations of binary Lennard-Jones systems with three different potentials, we show that the increase of anharmonicity and capacity for intermolecular coupling of the potential is the cause of (i) the increase of kinetic fragility and nonexponentiality in the liquid state, and (ii) the $T_g$-scaled temperature dependence of the nonergodicity parameter determined by the vibrations at low temperatures in the glassy state. Naturally, these parameters correlate with each other, as observed experimentally by T. Scopigno et al. [ Science 302, 849 (2003)].