We have used a combination of neutron scattering experiments and Monte Carlo simulations to study the initial stages of first-order phase transitions. We focus on quenches wherein the nascent phase is formed by homogeneous nucleation, and we approach the spinodal, i.e., the quench depth at which the original phase becomes unstable. In this regime, we show how critical nuclei sizes are determined from neutron scattering structure factors. Prevailing thought is that the size of the critical nucleus should increase with increasing quench depth and diverge at the spinodal. To the contrary, our experiments and simulations indicate that the critical nucleus size decreases monotonically as quench depth is increased and is finite at the spinodal.