The averaged null energy condition has been recently shown to hold for linear quantum fields in a large class of spacetimes. Nevertheless, it is easy to show by using a simple scaling argument that ANEC as stated cannot hold generically in curved four-dimensional spacetime, and this scaling argument has been widely interpreted as a death-blow for averaged energy conditions in quantum field theory. In this note I propose a simple generalization of ANEC, in which the right-hand-side of the ANEC inequality is replaced by a finite (but in general negative) state-independent lower bound. As long as attention is focused on asymptotically well-behaved spacetimes, this generalized version of ANEC is safe from the threat of the scaling argument, and thus stands a chance of being generally valid in four-dimensional curved spacetime. I argue that when generalized ANEC holds, it has implications for the non-negativity of total energy and for singularity theorems similar to the implications of ANEC. In particular, I show that if generalized ANEC is satisfied in static traversable wormhole spacetimes (which is likely but remains to be shown), then macroscopic wormholes (but not necessarily microscopic, Planck-size wormholes) are ruled out by quantum field theory.