This paper considers identifiability and recoverability in bilinear inverse problems which is relevant to blind deconvolution and matrix factorization. It is shown that bilinear inverse problems can be posed as rank-1 matrix recovery problems subject to linear constraints. Sufficient conditions for identifiability are developed for the cases when rank-2 matrices are present in the null space of the linear operator. Signal recovery using the nuclear norm heuristic for rank-1 matrix recovery is considered and simple conditions for success are provided.