Self-supervised contrastive learning is a powerful tool to learn visual representation without labels. Prior work has primarily focused on evaluating the recognition accuracy of various pre-training algorithms, but has overlooked other behavioral aspects. In addition to accuracy, distributional robustness plays a critical role in the reliability of machine learning models. We design and conduct a series of robustness tests to quantify the behavioral differences between contrastive learning and supervised learning to downstream or pre-training data distribution changes. These tests leverage data corruptions at multiple levels, ranging from pixel-level gamma distortion to patch-level shuffling and to dataset-level distribution shift. Our tests unveil intriguing robustness behaviors of contrastive and supervised learning. On the one hand, under downstream corruptions, we generally observe that contrastive learning is surprisingly more robust than supervised learning. On the other hand, under pre-training corruptions, we find contrastive learning vulnerable to patch shuffling and pixel intensity change, yet less sensitive to dataset-level distribution change. We attempt to explain these results through the role of data augmentation and feature space properties. Our insight has implications in improving the downstream robustness of supervised learning.