This paper considers the distribution of the cross-correlation sidelobes from intra-system GNSS interference after non-coherent integration. Because of the inherent periodic structure of the GNSS signals, the successive coherent integrations, while Gaussian distributed, are not independent. We model this dependence in the two separate cases of C/A and L5 signals. We then use the theory of Gaussian quadratic forms to specify the non-coherent integration distribution. Doppler and data bit combinations are shown to have an impact on the form of the distribution. Finally, we consider the special case of a single high-powered interferer. In this case, we use combinatorial reasoning to derive a distribution that is a mixture of all possible distributions for different values of the data bits.