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Models for predicting the time of a future event are crucial for risk assessment, across a diverse range of applications. Existing time-to-event (survival) models have focused primarily on preserving pairwise ordering of estimated event times (i.e., relative risk). We propose neural time-to-event models that account for calibration and uncertainty while predicting accurate absolute event times. Specifically, an adversarial nonparametric model is introduced for estimating matched time-to-event distributions for probabilistically concentrated and accurate predictions. We also consider replacing the discriminator of the adversarial nonparametric model with a survival-function matching estimator that accounts for model calibration. The proposed estimator can be used as a means of estimating and comparing conditional survival distributions while accounting for the predictive uncertainty of probabilistic models. Extensive experiments show that the distribution matching methods outperform existing approaches in terms of both calibration and concentration of time-to-event distributions.