We begin by examining the limitations of precedent-based explanations of the predicted outcome in case-based reasoning (CBR) approaches to classification and diagnosis. By failing to distinguish between features that support and oppose the predicted outcome, we argue, such explanations are not only less informative than might be expected, but also potentially misleading. To address this issue, we present an evidential approach to explanation in which a key role is played by techniques for the discovery of features that support or oppose the predicted outcome. Often in assessing the evidence provided by a continuous attribute, the problem is where to draw the line between values that support and oppose the predicted outcome. Our approach to the selection of such an evidence threshold is based on the weights of evidence provided by values above and below the threshold. Examples used to illustrate our evidential approach to explanation include a prototype CBR system for predicting whether or not a person is over the legal blood alcohol limit for driving based on attributes such as units of alcohol consumed.