The assumption of normality is crucial in many multivariate inference methods and may be even more important when the dimension of data is proportional to the sample size. It is therefore necessary that tests for multivariate non normality remain well behaved in such settings. In this article, we examine the properties of three common moment-based tests for non normality under increasing dimension asymptotics (IDA). It is demonstrated through Monte Carlo simulations that one of the tests is inconsistent under IDA and that one of them stands out as uniformly superior to the other two.