Generalized diversity combining (GDC), also known as hybrid selection/maximal ratio combining or generalized selection combining, is a low-complexity diversity combining technique by which a fixed subset of a large number of available diversity channels is chosen and then combined using the rules of maximal ratio combining. In this paper, we analyze the performance of GDC on time-correlated Rayleigh fading channels with noisy channel estimates. We derive expressions for the probability of error for various linear modulation schemes with coherent detection, and discuss the conditions under which the analysis can be extended to noncoherent and differentially coherent receiver structures. Throughout the paper, using a fundamental approach to obtain the decision statistic at the combiner output, a number of new expressions for the error probabilities are obtained in a rigorous way, along with a presentation of their performance with channel estimation errors. The final expressions have roughly the same complexity of evaluation as that for the channel with only additive Gaussian noise. Our results correct various inaccuracies in the literature, and show that coherent receivers based on imperfectly estimated channel knowledge incur a significant performance loss.