We consider the finite-sample power of various tests against serial correlation in the disturbances of a linear regression model when these disturbances follow certain stationary long-memory processes. It emerges that the power depends on the form of the regressor matrix and that, for the Durbin–Watson test and many other tests that can be written as ratios of quadratic forms in the disturbances, the power can drop to zero as the long-memory parameter approaches the boundary of the stationarity region. The problem does not arise when the regression includes an intercept. We also provide a means to detect this zero-power trap for given regressors. Our analytical results are illustrated using fractionally integrated white noise and ARFIMA(1, d, 0) disturbances with artificial regressors and with a real data set.