Testing the global earthquake catalogue for indications of non-Poissonian attributes has been an area of intense research, especially since the 2011 Tohoku earthquake. The usual approach is to test statistically the hypothesis that the global earthquake catalogue is well explained by a Poissonian process. In this paper we analyse one aspect of this problem which has been disregarded by the literature: the power of such tests to detect non-Poissonian features if they exist; that is, the probability of type II statistical errors. We argue in this article that the low frequency of large events and the brevity of our earthquake catalogues reduce the power of the statistical tests and render them unable to provide an unequivocal answer to this question. We do this using a counter example of a stochastic process that is clustered by construction and by analysing the resulting distribution of p-values given by a selection of statistical tests related to those found in the literature.