There exist a variety of reasons for the failure to find a unique cointegrating relationship between economic time series where one would normally be expected on the basis of economic theory. Among these are the testing procedure, the span of the data set, the choice of lag length in generating the test statistic, the presence of structural breaks, and the presence of cointegration only beyond some threshold. We propose the concept of regime-sensitive cointegration whereby the underlying series need not be cointegrated at all times. We show that cointegration can be switched off when a common stochastic trend is added. Alternatively, cointegration can be switched on or off because series normally believed to contain a unit actually do not. This implies that a linear combination of such variables need not be cointegrated. To illustrate the concept empirically, we test the hypothesis of interest-rate parity, and related hypotheses, using daily Eurorates for the United States and Canada.