Perceptual distances between images, as measured in the space of pre-trained deep features, have outperformed prior low-level, pixel-based metrics on assessing image similarity. While the capabilities of older and less accurate models such as AlexNet and VGG to capture perceptual similarity are well known, modern and more accurate models are less studied. First, we observe a surprising inverse correlation between ImageNet accuracy and Perceptual Scores of modern networks such as ResNets, EfficientNets, and Vision Transformers: that is better classifiers achieve worse Perceptual Scores. Then, we perform a large-scale study and examine the ImageNet accuracy/Perceptual Score relationship on varying the depth, width, number of training steps, weight decay, label smoothing, and dropout. Higher accuracy improves Perceptual Score up to a certain point, but we uncover a Pareto frontier between accuracies and Perceptual Score in the mid-to-high accuracy regime. We explore this relationship further using distortion invariance, spatial frequency sensitivity, and alternative perceptual functions. Interestingly we discover shallow ResNets, trained for less than 5 epochs only on ImageNet, whose emergent Perceptual Score matches the prior best networks trained directly on supervised human perceptual judgements.