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This paper examines the ability of greedy algorithms to estimate a block sparse parameter vector from noisy measurements. In particular, block sparse versions of the orthogonal matching pursuit and thresholding algorithms are analyzed under both adversarial and Gaussian noise models. In the adversarial setting, it is shown that estimation accuracy comes within a constant factor of the noise power. Under Gaussian noise, the Cramér-Rao bound is derived, and it is shown that the greedy techniques come close to this bound at high signal-to-noise ratio. The guarantees are numerically compared with the actual performance of block and non-block algorithms, identifying situations in which block sparse techniques improve upon the scalar sparsity approach. Specifically, we show that block sparse methods are particularly successful when the atoms within each block are nearly orthogonal.