While classical measurement error in the dependent variable in a linear regression framework results only in a loss of precision, non-classical measurement error can lead to estimates which are biased and inference which lacks power. Here, we consider a particular type of non-classical measurement error: skewed errors. Unfortunately, skewed measurement error is likely to be a relatively common feature of many outcomes of interest in political science research. This study highlights the bias that can result even from relatively "small" amounts of skewed measurement error, particularly if the measurement error is heteroskedastic. We also assess potential solutions to this problem, focusing on the stochastic frontier model and nonlinear least squares. Simulations and three replications highlight the importance of thinking carefully about skewed measurement error, as well as appropriate solutions.