Vision models notoriously flicker when applied to videos: they correctly recognize objects in some frames, but fail on perceptually similar, nearby frames. In this work, we systematically analyze the robustness of image classifiers to such temporal perturbations in videos. To do so, we construct two new datasets, ImageNet-Vid-Robust and YTBB-Robust, containing a total of 57,897 images grouped into 3,139 sets of perceptually similar images. Our datasets were derived from ImageNet-Vid and Youtube-BB, respectively, and thoroughly re-annotated by human experts for image similarity. We evaluate a diverse array of classifiers pre-trained on ImageNet and show a median classification accuracy drop of 16 and 10 points, respectively, on our two datasets. Additionally, we evaluate three detection models and show that natural perturbations induce both classification as well as localization errors, leading to a median drop in detection mAP of 14 points. Our analysis demonstrates that perturbations occurring naturally in videos pose a substantial and realistic challenge to deploying convolutional neural networks in environments that require both reliable and low-latency predictions.