Standard training techniques for neural networks involve multiple sources of randomness, e.g., initialization, mini-batch ordering and in some cases data augmentation. Given that neural networks are heavily over-parameterized in practice, such randomness can cause {\em churn} -- for the same input, disagreements between predictions of the two models independently trained by the same algorithm, contributing to the `reproducibility challenges' in modern machine learning. In this paper, we study this problem of churn, identify factors that cause it, and propose two simple means of mitigating it. We first demonstrate that churn is indeed an issue, even for standard image classification tasks (CIFAR and ImageNet), and study the role of the different sources of training randomness that cause churn. By analyzing the relationship between churn and prediction confidences, we pursue an approach with two components for churn reduction. First, we propose using \emph{minimum entropy regularizers} to increase prediction confidences. Second, \changes{we present a novel variant of co-distillation approach~\citep{anil2018large} to increase model agreement and reduce churn}. We present empirical results showing the effectiveness of both techniques in reducing churn while improving the accuracy of the underlying model.