Much research effort has been devoted to better understanding adversarial examples, which are specially crafted inputs to machine-learning models that are perceptually similar to benign inputs, but are classified differently (i.e., misclassified). Both algorithms that create adversarial examples and strategies for defending against them typically use $L_p$-norms to measure the perceptual similarity between an adversarial input and its benign original. Prior work has already shown, however, that two images need not be close to each other as measured by an $L_p$-norm to be perceptually similar. In this work, we show that nearness according to an $L_p$-norm is not just unnecessary for perceptual similarity, but is also insufficient. Specifically, focusing on datasets (CIFAR10 and MNIST), $L_p$-norms, and thresholds used in prior work, we show through online user studies that "adversarial examples" that are closer to their benign counterparts than required by commonly used $L_p$-norm thresholds can nevertheless be perceptually different to humans from the corresponding benign examples. Namely, the perceptual distance between two images that are "near" each other according to an $L_p$-norm can be high enough that participants frequently classify the two images as representing different objects or digits. Combined with prior work, we thus demonstrate that nearness of inputs as measured by $L_p$-norms is neither necessary nor sufficient for perceptual similarity, which has implications for both creating and defending against adversarial examples. We propose and discuss alternative similarity metrics to stimulate future research in the area.