Tax brackets are a common feature of non-renewable resource taxes. Although the introduction of brackets would seem to be an innocuous way of approximating a non-linear tax system, the dynamic effects are surprising. This paper shows that the presence of tax brackets in both a severance tax system and a profits tax system can induce the extractive firm to depart from the well-known monotonically declining extraction profile and to choose instead a profile which has constant extraction rates over some interval of time.