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This paper aims to improve video streaming by leveraging a simple observation: users are more sensitive to low quality in certain parts of a video than in others. For instance, rebuffering during key moments of a sports video (e.g., before a goal is scored) is more annoying than rebuffering during normal gameplay. Such dynamic quality sensitivity, however, is rarely captured by current approaches, which predict QoE (quality-of-experience) using one-size-fits-all heuristics that are too simplistic to understand the nuances of video content. Instead of proposing yet another heuristic, we take a different approach: we run a separate crowdsourcing experiment for each video to derive users' quality sensitivity at different parts of the video. Of course, the cost of doing this at scale can be prohibitive, but we show that careful experiment design combined with a suite of pruning techniques can make the cost negligible compared to how much content providers invest in content generation and distribution. Our ability to accurately profile time-varying user sensitivity inspires a new approach: dynamically aligning higher (lower) quality with higher (lower) sensitivity periods. We present a new video streaming system called SENSEI that incorporates dynamic quality sensitivity into existing quality adaptation algorithms. We apply SENSEI to two state-of-the-art adaptation algorithms. SENSEI can take seemingly unusual actions: e.g., lowering bitrate (or initiating a rebuffering event) even when bandwidth is sufficient so that it can maintain a higher bitrate without rebuffering when quality sensitivity becomes higher in the near future. Compared to state-of-the-art approaches, SENSEI improves QoE by 15.1% or achieves the same QoE with 26.8% less bandwidth on average.