Spin-polarized gapless surface states in topological insulators form chiral Dirac cones. When such materials are reduced to thin films, the Dirac states on the two faces of the film can overlap and couple by quantum tunneling, resulting in a thickness-dependent insulating gap at the Dirac point. Calculations for a freestanding Sb film with a thickness of four atomic bilayers yield a gap of 36 meV, yet angle-resolved photoemission measurements of a film grown on Si(111) reveal no gap formation. The surprisingly robust Dirac cone is explained by calculations in terms of interfacial interaction.