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The distribution of the aggregate interference power in large wireless networks has gained increasing attention with the emergence of different types of wireless networks such as ad hoc networks, sensor networks, and cognitive radio networks. The interference in such networks is often characterized using the Poisson point process (PPP). As the number of interfering nodes increases, there might be a tendency to approximate the distribution of the aggregate interference power by a Gaussian random variable, given that the individual interference signals are independent. However, some observations in the literature suggest that this Gaussian approximation is not valid, except under some specific scenarios. In this paper, we cast these observations in a single mathematical framework and express the conditions for which the Gaussian approximation will be valid for the aggregate interference power generated by a Poisson field of interferers. Furthermore, we discuss the effect of different system and channel parameters on the convergence of the distribution of the aggregate interference to a Gaussian distribution.