In this paper, the robust H-infinity state estimation problem is investigated for a general class of uncertain discrete-time stochastic neural networks with probabilistic measurement delays. The measurement delays of the neural networks are described by a binary switching sequence satisfying a conditional probability distribution. The neural network under study involves parameter uncertainties, stochastic disturbances and time-varying delays, and the activation functions are characterized by sector-like nonlinearities. The problem addressed is the design of a full-order state estimator, for all admissible uncertainties, nonlinearities and time-delays, the dynamics of the estimation error is constrained to be robustly exponentially stable in the mean square and, at the same time, a prescribed H1 disturbance rejection attenuation level is guaranteed. By using the Lyapunov stability theory and stochastic analysis techniques, sufficient conditions are first established to ensure the existence of the desired estimators. These conditions are dependent on the lower and upper bounds of the time-varying delays. Then, the explicit expression of the desired estimator gains is described in terms of the solution to a linear matrix inequality (LMI). Finally, a numerical example is exploited to show the usefulness of the results derived.