Both quantum mechanics and Markov-process theory deal with dynamical systems whose behaviors can be predicted only probabilistically. The question therefore arises as to whether Markov-process theory, a theory that is firmly rooted in the common-sense laws of probability, can provide a generally adequate mathematical framework for quantum mechanics, a theory whose predictions often seem to defy common sense. It is shown here that the measurable behavior of a certain elementary quantum system that oscillates between two discrete states cannot be modeled as a Markov process. That this will also be true for most other quantum systems is shown to follow from the same property of quantum mechanics that is responsible for the so-called quantum Zeno effect. The conclusion that quantum behavior cannot generally be modeled as a Markov process underscores the truly mysterious nature of quantum mechanics when viewed in common-sense terms.