How reliably can we trust the scores obtained from social bias benchmarks as faithful indicators of problematic social biases in a given language model? In this work, we study this question by contrasting social biases with non-social biases stemming from choices made during dataset construction that might not even be discernible to the human eye. To do so, we empirically simulate various alternative constructions for a given benchmark based on innocuous modifications (such as paraphrasing or random-sampling) that maintain the essence of their social bias. On two well-known social bias benchmarks (Winogender and BiasNLI) we observe that these shallow modifications have a surprising effect on the resulting degree of bias across various models. We hope these troubling observations motivate more robust measures of social biases.