Vision transformers (ViTs) have recently set off a new wave in neural architecture design thanks to their record-breaking performance in various vision tasks. In parallel, to fulfill the goal of deploying ViTs into real-world vision applications, their robustness against potential malicious attacks has gained increasing attention. In particular, recent works show that ViTs are more robust against adversarial attacks as compared with convolutional neural networks (CNNs), and conjecture that this is because ViTs focus more on capturing global interactions among different input/feature patches, leading to their improved robustness to local perturbations imposed by adversarial attacks. In this work, we ask an intriguing question: "Under what kinds of perturbations do ViTs become more vulnerable learners compared to CNNs?" Driven by this question, we first conduct a comprehensive experiment regarding the robustness of both ViTs and CNNs under various existing adversarial attacks to understand the underlying reason favoring their robustness. Based on the drawn insights, we then propose a dedicated attack framework, dubbed Patch-Fool, that fools the self-attention mechanism by attacking its basic component (i.e., a single patch) with a series of attention-aware optimization techniques. Interestingly, our Patch-Fool framework shows for the first time that ViTs are not necessarily more robust than CNNs against adversarial perturbations. In particular, we find that ViTs are more vulnerable learners compared with CNNs against our Patch-Fool attack which is consistent across extensive experiments, and the observations from Sparse/Mild Patch-Fool, two variants of Patch-Fool, indicate an intriguing insight that the perturbation density and strength on each patch seem to be the key factors that influence the robustness ranking between ViTs and CNNs. It can be expected that our Patch-Fool framework will shed light on both future architecture designs and training schemes for robustifying ViTs towards their real-world deployment. Our codes are available at https://github.com/RICE-EIC/Patch-Fool.