In order to reliably process natural language, NLP systems must generalize to the long tail of rare utterances. We propose a method to create challenging benchmarks that require generalizing to the tail of the distribution by re-splitting existing datasets. We create ‘Likelihood splits’ where examples that are assigned lower likelihood by a pre-trained language model (LM) are placed in the test set, and more likely examples are in the training set. This simple approach can be customized to construct meaningful train-test splits for a wide range of tasks. Likelihood splits are more challenging than random splits: relative error rates of state-of-the-art models on our splits increase by 59% for semantic parsing on Spider, 77% for natural language inference on SNLI, and 38% for yes/no question answering on BoolQ compared with the corresponding random splits. Moreover, Likelihood splits create fairer benchmarks than adversarial filtering; when the LM used to create the splits is used as the task model, our splits do not adversely penalize the LM.