It is well known that, keeping fixed the non-centrality parameter, the non-central Student’s t distribution tends to the Gaussian distribution when the degrees of freedom diverge. In this paper, it is shown that, under certain conditions, an analogous result holds also when the degrees of freedom and the non-centrality parameter diverge together. Further, the connection between the parameters of the limiting Gaussian distribution and the growth rates of the non-centrality parameter is exploited.