The counterintuitive phenomenon of pressure-induced softening in materials is likely to be caused by the same dynamical behavior that produces negative thermal expansion. Through a combination of molecular dynamics simulation on an idealized model and neutron diffraction at variable temperature and pressure, we show the existence of extraordinary and unprecedented pressure-induced softening in the negative thermal expansion material scandium fluoride ScF_{3}. The pressure derivative of the bulk modulus B, B^{'}=(∂B/∂P)_{P=0}, reaches values as low as -220±30 at 50 K, and is constant at -50 between 150 and 250 K.