A Bayesian perspective on Ioannidis’s (2005) memorable statement that “Most Published Research Findings Are False” suggests a seemingly inescapable trade-off: It appears as if research hypotheses are based either on safe ground (high prior odds), yielding valid but unsurprising results, or on unexpected and novel ideas (low prior odds), inspiring risky and surprising findings that are inevitably often wrong. Indeed, research of two prominent types, sexy hypothesis testing and model testing, is often characterized by low priors (due to astounding hypotheses and conjunctive models) as well as low-likelihood ratios (due to nondiagnostic predictions of the yin-or-yang type). However, the trade-off is not inescapable: An alternative research approach, theory-driven cumulative science, aims at maximizing both prior odds and diagnostic hypothesis testing. The final discussion emphasizes the value of pluralistic science, within which exploratory phenomenon-driven research can play a similarly strong part as strict theory-testing science.