Changing just a few words in a story can induce a substantial change in the overall narrative. How does the brain accumulate and process local and sparse changes, creating a unique situation model of the story, over the course of a real-life narrative? Recently, we mapped a hierarchy of processing timescales in the brain: from early sensory areas that integrate information over 10s-100s ms, to high-order areas that integrate information over many seconds to minutes. Based on this hierarchy, we hypothesize that early sensory areas would be sensitive to local changes in word use, but that there will be increasingly divergent neural responses along the processing hierarchy as higher-order areas accumulate and amplify these local changes. To test this hypothesis, we created two structurally related but interpretively distinct narratives by changing some individual words. We found that the neural response distance between the stories was amplified as story information is transferred from low-level regions (e.g. early auditory cortex) to high-level regions (e.g precuneus and prefrontal cortex) and that the neural difference between stories is highly correlated with an area's ability to integrate information over time. Our results suggest a neural mechanism by which two similar situations become easy to distinguish.