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We bound the number of sensors required to achieve a desired level of sensing accuracy in a discrete sensor network application (e.g. distributed detection). We model the state of nature being sensed as a discrete vector, and the sensor network as an encoder. Our model assumes that each sensor observes only a subset of the state of nature, that sensor observations are localized and dependent, and that sensor network output across different states of nature is neither identical nor independently distributed. Using a random coding argument we prove a lower bound on the 'sensing capacity' of a sensor network, which characterizes the ability of a sensor network to distinguish among all states of nature. We compute this lower bound for sensors of varying range, noise models, and sensing functions. We compare this lower bound to the empirical performance of a belief propagation based sensor network decoder for a simple seismic sensor network scenario. The key contribution of this paper is to introduce the idea of a sharp cut-off function in the number of required sensors, to the sensor network community.