Scalars and fermions can arise as Goldstone modes of nonlinearly realized extensions of the Poincare group (with important implications for the soft limits of such theories): the Dirac-Born-Infeld scalar realizes a higher-dimensional Poincare symmetry, while the Volkov-Akulov fermion corresponds to super-Poincare. In this paper we classify extensions of the Poincare group which give rise to a vector Goldstone mode instead. Our main result is that there are no healthy (ghost free) interacting U(1) gauge theories that nonlinearly realize space-time symmetries beyond gauge transformations. This implies that the structure of e.g., Born-Infeld theory is not fixed by symmetry.