Scanning tunneling microscopy experiments reveal that small Pb islands with unstable heights, e.g., four layers, on a Si(111) surface decay during coarsening, whereas large islands do not decay but grow to a stable height. This bifurcation in evolution is analyzed by incorporating quantum size effects into theoretical models for island growth dynamics with appropriate geometries. The effective energy barrier for Pb atoms to reach the top of four-layer islands is estimated at about 0.26 eV.