Based on either Monte Carlo simulations, or several examples based on actual data, I show that the ability of Johansen's tests to detect a cointegration relationship significantly deteriorates under two empirically plausible circumstances: (i ) when, in addition to a cointegration relationship, a system features one or more "nuisance" series - i.e., series driven by permanent shocks different from those driving the cointegration relationship; and (ii ) when a system features multiple cointegration relationships driven by different permanent shocks, as implied (e.g.) by the Classical Dichotomy (this being a special case of (i )). These results suggest that performing Johansen's tests based on systems featuring both real and nominal series automatically biases the tests against rejecting the null. The substantive implication for applied research is that, when searching for cointegration based on Johansen's tests, a cointegration relationship should be tested based on the smallest system for which economic theory suggests cointegration should hold. I provide several illustrations of how failure to do so results in cointegration not being detected between (e.g.) either M1 velocity and a short-term rate; real house prices and real rents; GDP and consumption; or short- and long-term interest rates.