One way to interpret neural model predictions is to highlight the most important input features---for example, a heatmap visualization over the words in an input sentence. In existing interpretation methods for NLP, a word's importance is determined by either input perturbation---measuring the decrease in model confidence when that word is removed---or by the gradient with respect to that word. To understand the limitations of these methods, we use input reduction, which iteratively removes the least important word from the input. This exposes pathological behaviors of neural models: the remaining words appear nonsensical to humans and are not the ones determined as important by interpretation methods. As we confirm with human experiments, the reduced examples lack information to support the prediction of any label, but models still make the same predictions with high confidence. To explain these counterintuitive results, we draw connections to adversarial examples and confidence calibration: pathological behaviors reveal difficulties in interpreting neural models trained with maximum likelihood. To mitigate their deficiencies, we fine-tune the models by encouraging high entropy outputs on reduced examples. Fine-tuned models become more interpretable under input reduction without accuracy loss on regular examples.