Existing algorithms for dictionary learning assume that the entries of the (high-dimensional) input data are fully observed. However, in several practical applications, only an incomplete fraction of the data entries may be available. For incomplete settings, no provably correct and polynomialtime algorithm has been reported in the dictionary learning literature. In this paper, we provide provable approaches for learning - from incomplete samples - a family of dictionaries whose atoms have sufficiently "spread-out" mass. First, we propose a descent-style iterative algorithm that linearly converges to the true dictionary when provided a sufficiently coarse initial estimate. Second, we propose an initialization algorithm that utilizes a small number of extra fully observed samples to produce such a coarse initial estimate. Finally, we theoretically analyze their performance and provide asymptotic statistical and computational guarantees.