Recently, a groundswell of research has identified the use of counterfactual explanations as a potentially significant solution to the Explainable AI (XAI) problem. It is argued that (a) technically, these counterfactual cases can be generated by permuting problem-features until a class change is found, (b) psychologically, they are much more causally informative than factual explanations, (c) legally, they are GDPR-compliant... However, there are issues around the finding of good counterfactuals using current techniques (e.g. sparsity and plausibility). We show that many commonly-used datasets appear to have few good counterfactuals for explanation purposes. So, we propose a new case based approach for generating counterfactuals using novel ideas about the counterfactual potential and explanatory coverage of a case-base. The new technique reuses patterns of good counterfactuals, present in a case-base, to generate analogous counterfactuals that can explain new problems and their solutions. Several experiments show how this technique can improve the counterfactual potential and explanatory coverage of case-bases that were previously found wanting. (read more)