To increase trust in artificial intelligence systems, a promising research direction consists of designing neural models capable of generating natural language explanations for their predictions. In this work, we show that such models are nonetheless prone to generating mutually inconsistent explanations, such as "Because there is a dog in the image" and "Because there is no dog in the [same] image", exposing flaws in either the decision-making process of the model or in the generation of the explanations. We introduce a simple yet effective adversarial framework for sanity checking models against the generation of inconsistent natural language explanations. Moreover, as part of the framework, we address the problem of adversarial attacks with full target sequences, a scenario that was not previously addressed in sequence-to-sequence attacks. Finally, we apply our framework on a state-of-the-art neural natural language inference model that provides natural language explanations for its predictions. Our framework shows that this model is capable of generating a significant number of inconsistent explanations.