As discussed by Perron (1989), a common problem when testing for unit roots is the presence of a structural break that has not been accounted for in the testing procedure. In such cases, unit root tests are biased to non-rejection of the null hypothesis of non-stationarity. These results have been discussed using asymptotic theory and large samples in papers by Leybourne and Newbold (2000), Montanes and Reyes (1998) and Lee, Huang, and Shin (1997). In this paper we investigate the impact of ignoring structural breaks on sample sizes of more interest to empirical economists and show the results on power and size for both tests of the null of non-stationarity (ADF and Phillips-Perron) and the null of stationarity (KPSS). We are also able to give some guidelines on break placement which can cause the rapid flipping of rejection probabilities as discussed in Leybourne and Newbold (2000). Finally, we provide examples from time series data in South Africa to show the danger of misdiagnosis and the resulting misspecifications that can occur.