When can a Bayesian select an hypothesis H and design an experiment (or a sequence of experiments) to make certain that, given the experimental outcome(s), the posterior probability of H will be greater than its prior probability? We discuss an elementary result which establishes sufficient conditions under which this cannot occur. We illustrate how, when the sufficient conditions fail, because probability is finitely but not countably additive, it may be that a Bayesian can design an experiment to lead his/her posterior probability into a foregone conclusion. The problem has a decision theoretic version, which we discuss from several perspectives. Also we relate this issue in Bayesian hypothesis testing to various concerns about "optimal stopping."