On some accounts of vagueness, predicates like “is a heap” are tolerant. That is, their correct application tolerates sufficiently small changes in the objects to which they are applied. (So, according to tolerant views, if a given object is a heap, it will necessarily remain a heap after one grain of sand is removed.) Of course, such views face the sorites paradox, and various solutions have been proposed. One proposed solution involves banning repeated appeals to tolerance, while affirming tolerance in any individual case. (So, you may always remove one grain of sand safely, but you mustn’t make a habit of it.) In effect, this solution rejects the reasoning (rather than the premises) of the sorites argument. This paper discusses a thorny problem afflicting this approach to vagueness. In particular, it is shown that, on the foregoing view, whether an object is a heap will sometimes depend on factors extrinsic to that object, such as whether its components came from other heaps. More generally, the paper raises the issue of how to count heaps in a tolerance-friendly framework.