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Recently, transformer-based methods such as RoBERTa and GPT-3 have led to significant experimental advances in natural language processing tasks such as question answering and commonsense reasoning. The latter is typically evaluated through multiple benchmarks framed as multiple-choice instances of the former. According to influential leaderboards hosted by the Allen Institute (evaluating state-of-the-art performance on commonsense reasoning benchmarks), models based on such transformer methods are approaching human-like performance and have average accuracy well over 80% on many benchmarks. Since these are commonsense benchmarks, a model that generalizes on commonsense reasoning should not experience much performance loss across multiple commonsense benchmarks. In this paper, we study the generalization issue in detail by designing and conducting a rigorous scientific study. Using five common benchmarks, multiple controls and statistical analysis, we find clear evidence that fine-tuned commonsense language models still do not generalize well, even with moderate changes to the experimental setup, and may, in fact, be susceptible to dataset bias. We also perform selective studies, including qualitative and consistency analyses, to gain deeper insight into the problem.