Standard target zone exchange rate models are based on nonlinear functions of an unobserved economic fundamental, which is assumed to be bounded, similarly to the target zone exchange rates themselves. A violation of this key assumption is a basic structural reason for model failure. Using a novel estimation and testing strategy, we show it is also a testable assumption. Our empirical results cast serious doubt on its validity in practice, providing a primitive reason for well-documented rejections of the basic model. Model failure from this violation is robust to otherwise ideal circumstances (e.g., perfect credibility)