Random connection models form a family of random graph models. In our case the vertex set is distributed as a Poisson point process on hyperbolic d-space, and an edge exists between vertices x and y independently with a probability that depends only on the (hyperbolic) distance between them. Under very mild conditions, these models exhibit a critical intensity (for the Poisson point process) such that all connected components of the graph are almost surely finite for intensities below this critical intensity, and there almost surely exists an infinite connected component for intensities above this critical intensity. In this talk we will consider the number of infinite connected components in this super-critical phase, and discuss the consequences of this argument for various aspects of mean-field behaviour in this percolation model.