Current pre-trained language models have enabled remarkable improvements in downstream tasks, but it remains difficult to distinguish effects of statistical correlation from more systematic logical reasoning grounded on understanding of the real world. In this paper we tease these factors apart by leveraging counterfactual conditionals, which force language models to predict unusual consequences based on hypothetical propositions. We introduce a set of tests drawn from psycholinguistic experiments, as well as larger-scale controlled datasets, to probe counterfactual predictions from a variety of popular pre-trained language models. We find that models are consistently able to override real-world knowledge in counterfactual scenarios, and that this effect is more robust in case of stronger baseline world knowledge---however, we also find that for most models this effect appears largely to be driven by simple lexical cues. When we mitigate effects of both world knowledge and lexical cues to test knowledge of linguistic nuances of counterfactuals, we find that only GPT-3 shows sensitivity to these nuances, though this sensitivity is also non-trivially impacted by lexical associative factors.