There has been tremendous progress in generating realistic faces with high fidelity over the past few years. Despite this progress, a crucial question remains unanswered: "Given a generative face model, how many unique identities can it generate?" In other words, what is the biometric capacity of the generative face model? A scientific basis for answering this question will benefit evaluating and comparing different generative face models and establish an upper bound on their scalability. This paper proposes a statistical approach to estimate the biometric capacity of generated face images in a hyperspherical feature space. We employ our approach on multiple generative models, including unconditional generators like StyleGAN, Latent Diffusion Model, and "Generated Photos," as well as DCFace, a class-conditional generator. We also estimate capacity w.r.t. demographic attributes such as gender and age. Our capacity estimates indicate that (a) under ArcFace representation at a false acceptance rate (FAR) of 0.1%, StyleGAN3 and DCFace have a capacity upper bound of 1.43 \times 10^{6} and 1.190 \times 10^{4}, respectively; (b) the capacity reduces drastically as we lower the desired FAR with an estimate of 1.796 \times 10^{4} and 562 at FAR of 1% and 10%, respectively, for StyleGAN3; (c) there is no discernible disparity in the capacity w.r.t gender; and (d) for some generative models, there is an appreciable disparity in the capacity w.r.t age. Code is available at https://github.com/humananalysis/capacity-generative-face-models.