Can Theta/N Dependence for Gluodynamics be Compatible with 2 pi Periodicity in Theta ?

In a number of field theoretical models the vacuum angle \theta enters physics in the combination \theta/N, where N stands generically for the number of colors or flavors, in an apparent contradiction with the expected 2 \pi periodicity in \theta. We argue that a resolution of this puzzle is related to the existence of a number of different \theta dependent sectors in a finite volume formulation, which can not be seen in the naive thermodynamic limit V -> \infty. It is shown that, when the limit V -> \infty is properly defined, physics is always 2 \pi periodic in \theta for any integer, and even rational, values of N, with vacuum doubling at certain values of \theta. We demonstrate this phenomenon in both the multi-flavor Schwinger model with the bosonization technique, and four-dimensional gluodynamics with the effective Lagrangian method. The proposed mechanism works for an arbitrary gauge group.