We give a polynomial-time algorithm for provably learning the structure and parameters of bipartite noisy-or Bayesian networks of binary variables where the top layer is completely hidden. Unsupervised learning of these models is a form of discrete factor analysis, enabling the discovery of hidden variables and their causal relationships with observed data. We obtain an efficient learning algorithm for a family of Bayesian networks that we call quartet-learnable, meaning that every latent variable has four children that do not have any other parents in common. We show that the existence of such a quartet allows us to uniquely identify each latent variable and to learn all parameters involving that latent variable. Underlying our algorithm are two new techniques for structure learning: a quartet test to determine whether a set of binary variables are singly coupled, and a conditional mutual information test that we use to learn parameters. We also show how to subtract already learned latent variables from the model to create new singly-coupled quartets, which substantially expands the class of structures that we can learn. Finally, we give a proof of the polynomial sample complexity of our learning algorithm, and experimentally compare it to variational EM.