A matrix is blocky if it is a "blowup" of a permutation matrix. The blocky rank of a matrix M is the minimum number of blocky matrices that linearly span M. Hambardzumyan, Hatami and Hatami defined blocky rank and showed that it is connected to communication complexity and operator theory. We describe additional connections to circuit complexity and combinatorics, and we prove upper and lower bounds on blocky rank in various contexts.