We develop a framework that we call compressive rate estimation. We assume that the composite channel gain matrix (i.e. the matrix of all channel gains between all network nodes) is compressible which means it can be approximated by a sparse or low rank representation. We develop and study a novel sensing and reconstruction protocol for the estimation of achievable rates. We develop a sensing protocol that exploits the superposition principle of the wireless channel and enables the receiving nodes to obtain non-adaptive random measurements of columns of the composite channel matrix. The random measurements are fed back to a central controller that decodes the composite channel gain matrix (or parts of it) and estimates individual user rates. We analyze the rate loss for a linear and a non-linear decoder and find the scaling laws according to the number of non-adaptive measurements. In particular if we consider a system with $N$ nodes and assume that each column of the composite channel matrix is $k$ sparse, our findings can be summarized as follows. For a certain class of non-linear decoders we show that if the number of pilot signals $M$ scales like $M \sim k \log(N/k)$, then the rate loss compared to perfect channel state information remains bounded. For a certain class of linear decoders we show that the rate loss compared to perfect channel state information scales like $1/\sqrt{M}$.