This paper demonstrates that unit root tests can suffer from inflated Type I error rates when data are cointegrated. Results from Monte Carlo simulations show that three commonly used unit root tests - the ADF, Phillips-Perron, and DF-GLS tests - frequently overreject the true null of a unit root for at least one of the cointegrated variables. The reason for this overrejection is that unit root tests, designed for random walk data, are often misspecified when data are cointegrated. While the addition of lagged differenced (LD) terms can eliminate the size distortion, this "success" is spurious, driven by collinearity between the lagged dependent variable and the LD explanatory variables. Accordingly, standard diagnostics such as (i) testing for serial correlation in the residuals and (ii) using information criteria to select among different lag specifications are futile. The implication of these results is that researchers should be conservative in the weight they attach to individual unit root tests when determining whether data are cointegrated.