The mean-square displacement (MSD) was measured by neutron scattering at various temperatures and pressures for a number of molecular glass-forming liquids. The MSD is invariant along the glass-transition line at the pressure studied, thus establishing an "intrinsic" Lindemann criterion for any given liquid. A one-to-one connection between the MSD's temperature dependence and the liquid's fragility is found when the MSD is evaluated on a time scale of ∼4 ns , but does not hold when the MSD is evaluated at shorter times. The findings are discussed in terms of the elastic model and the role of relaxations, and the correlations between slow and fast dynamics are addressed.