The realizability of Reynolds stress models in homogeneous turbulence is critically assessed from a theoretical standpoint. It is proven that a well known second-order closure model formulated using the strong realizability constraints is, in fact, not a realizable model. On the other hand, models properly formulated to satisfy the weak form of realizability -- wherein states of one or two component turbulence are made inaccessible in finite time via the imposition of a positive first derivative condition -- are found to be realizable.