In this letter, we study the effects that perception latency has on the maximum speed a robot can reach to safely navigate through an unknown cluttered environment. We provide a general analysis that can serve as a baseline for future quantitative reasoning for design tradeoffs in autonomous robot navigation. We consider the case where the robot is modeled as a linear secondorder system with bounded input and navigates through static obstacles. Also, we focus on a scenario where the robot wants to reach a target destination in as little time as possible, and therefore cannot change its longitudinal velocity to avoid obstacles. We show how the maximum latency that the robot can tolerate to guarantee safety is related to the desired speed, the range of its sensing pipeline, and the actuation limitations of the platform (i.e., the maximum acceleration it can produce). As a particular case study, we compare monocular and stereo frame-based cameras against novel, low-latency sensors, such as event cameras, in the case of quadrotor flight. To validate our analysis, we conduct experiments on a quadrotor platform equipped with an event camera to detect and avoid obstacles thrown towards the robot. To the best of our knowledge, this is the first theoretical work in which perception and actuation limitations are jointly considered to study the performance of a robotic platform in high-speed navigation.