How far from non-differential does exposure or disease misclassification have to be to bias measures of association away from the null?

A well-known heuristic in epidemiology is that non-differential exposure or disease misclassification biases the expected values of an estimator toward the null value. This heuristic works correctly only when additional conditions are met, such as independence of classification errors. We present examples to show that, even when the additional conditions are met, if the misclassification is only approximately non-differential, then bias is not guaranteed to be toward the null. In light of such examples, we advise that evaluation of misclassification should not be based on the assumption of exact non-differentiality unless the latter can be deduced logically from the facts of the situation.