Recently, convolutional neural networks (CNNs) have been proposed as a method for deformable image registration, offering a variety of potential advantages compared to physical model-based methods, including faster runtime and ability to learn complicated functions without explicit models. A persistent question for CNNs is the uncertainty in their behavior when the image statistics (e.g., noise and resolution) of the test data deviate from those of the training data. In this work we investigated the influence of statistical properties of image noise (in CT, for example, related to radiation dose) and deformation magnitude, trained registration networks over a range of dose and deformation levels, and evaluated registration performance (target registration error, TRE) as the statistics of the test data deviated from that of the training data. Generally, registration performance was optimal when the statistics of the test data matched that of the training data, except in cases of very low-dose data, where networks trained on a combination of high- and low-dose images achieved best TRE. Furthermore, TRE was found to be limited by the highest dose training data, with no improvement in TRE for test images of higher dose than that in the training data. Understanding and quantifying the relationship between statistical aspects of the training and test data - and the failure modes caused by statistical mismatch - is an important step in the development of CNN-based registration methods. This work provided new insight on the optima and tradeoffs with respect to image noise (dose) and deformation magnitude, providing important guidance in building training sets that are bestsuited to particular imaging conditions and applications.