Bridging NLP with political science, this paper examines both the potential and the limitations of a computational hate speech detection method in addressing real-world questions. Using Greece as a case study, we analyze over 4 million tweets from 2015 to 2022—a period marked by economic, refugee, foreign policy, and pandemic crises. The analysis of false positives highlights the challenges of accurately detecting different types of verbal attacks across various targets and timeframes. In addition, the analysis of true positives reveals distinct linguistic patterns that reinforce populist narratives, polarization and hostility. By situating these findings within their socio-political context, we provide insights into how hate speech manifests online in response to real-world crises.