In recent research, human-understandable explanations of machine learning models have received a lot of attention. Often explanations are given in form of model simplifications or visualizations. However, as shown in cognitive science as well as in early AI research, concept understanding can also be improved by the alignment of a given instance for a concept with a similar counterexample. Contrasting a given instance with a structurally similar example which does not belong to the concept highlights what characteristics are necessary for concept membership. Such near misses have been proposed by Winston (Learning structural descriptions from examples, 1970) as efficient guidance for learning in relational domains. We introduce an explanation generation algorithm for relational concepts learned with Inductive Logic Programming (GeNME). The algorithm identifies near miss examples from a given set of instances and ranks these examples by their degree of closeness to a specific positive instance. A modified rule which covers the near miss but not the original instance is given as an explanation. We illustrate GeNME with the well-known family domain consisting of kinship relations, the visual relational Winston arches domain, and a real-world domain dealing with file management. We also present a psychological experiment comparing human preferences of rule-based, example-based, and near miss explanations in the family and the arches domains.