We address the downlink channel estimation problem for massive multiple-input multiple-output (MIMO) systems in this paper, where the inherit burst-sparsity structure is exploited to improve the channel estimation performance. In the literature, the commonly used burst-sparsity model assumes a uniform burst-sparse structure in which all bursts have similar sizes. However, such assumption is oversimplified to hold in practice. Outliers deviated from such uniform burst structures can significantly degrade the accuracy of the existing burst-sparsity models, which may result in a reduced recovery performance. To capture a more general burst-sparsity structure in practice, we propose a novel non-uniform burst-sparsity model and introduce an improved pattern-coupled prior to account for more realistic non-uniform burst structures. A generic sparse Bayesian learning based framework to exploit the non-uniform burst-sparsity and to enhance massive MIMO channel estimation performance is then developed. We further prove that our solution converges to a stationary point of the associated optimization problem, and our framework includes the state-of-the-art pattern-coupled method as a special case. Simulation results verify the robust performance of the devised method.