Counterfactuals are a valuable means for understanding decisions made by ML systems. However, the counterfactuals generated by the methods currently available for natural language text are either unrealistic or introduce imperceptible changes. We propose CounterfactualGAN: a method that combines a conditional GAN and the embeddings of a pretrained BERT encoder to model-agnostically generate realistic natural language text counterfactuals for explaining regression and classification tasks. Experimental results show that our method produces perceptibly distinguishable counterfactuals, while outperforming four baseline methods on fidelity and human judgments of naturalness, across multiple datasets and multiple predictive models.