We demonstrate experimentally the existence of two stability regimes of Ag nanoislands grown on a Si(111)-(4 x 1)-In surface: a conventional regime at low temperature where only one island shape is stable, and an unconventional regime at room temperature (RT) where isotropic compact islands coexist with anisotropic elongated ones. First-principles calculations show the unusual bistability at RT arises from the fact that the Ag nanoislands are under anisotropic stress, supporting a recent theoretical prediction by Zandvliet and van Gastel [Phys. Rev. Lett. 99, 136103 (2007)10.1103/PhysRevLett.99.136103].