Longley [10] proposed a set of data for use in testing regression programs. This paper shows that the numerically accurate solution in this example was probably an unreasonable estimate of the regression coefficients. This is true because the accuracy of the data and appropriateness of the model may affect the solution more than the computational method. An easily computed index is derived that can be used to indicate such computational instability. The basic conclusion is that a concern about highly accurate computational methods must be tempered with a concern for whether the data are accurate enough to make the results meaningful.