Recent research has established the utility of adaptive bounds on model uncertainty in adaptive approximation-based control. Such bounds have utility both for robust control law design and for self-organizing approximators that could adjust the number of basis elements N by adding additional approximation resources in the regions where the approximation error bound is large. Existing adaptive bounding methods utilize algorithms with global forgetting. In this article, we investigate methods to develop bounds on approximation accuracy that involve local forgetting. The importance of local versus global forgetting is motivated in the text and illustrated with an example.