In this paper, we study the connectivity properties of a wireless sensor network from an information-theoretic viewpoint. We consider both regular linear (one-dimensional) and planar (two-dimensional) networks with unreliable sensor nodes, i.e., each node is inactive/dead with a certain probability. We study the following problems: 1) What is the fundamental limit on the data rate that such a network can support for a single sensor node to the destination under transmission power and network-topology constraints? 2) What are the constraints on the network topology such that any (single) sensor can communicate with the destination at a desired rate? For problem 1), we provide upper and lower bounds on the achievable data rate, and for problem 2), we provide upper and lower bounds on the distance between the nodes required for communication at the desired rate.