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When inferring the evolutionary history of species and the genes they contain, the phylogenetic trees of genes can be different from those of the species and to each other, due to a variety of causes, including incomplete lineage sorting. We often wish to infer the species tree, but only reconstruct the gene trees from sequences. We then combine the gene trees to produce a species tree; methods to do this are known as summary methods, of which ASTRAL is currently among the most popular. ASTRAL has been shown to be accurate in many practical scenarios through extensive simulations. However, these simulations generally assume that the input gene trees are independent of each other (infinite recombination between loci). This is known to be unrealistic, as genes that are close to each other on the chromosome (or are co-evolving) have dependent phylogenies. In this paper, we develop a model for generating dependent gene trees within a species tree, based on the coalescent with recombination. We then use these trees as input to ASTRAL to reassess its accuracy for dependent gene trees. Our results allow us to evaluate the impact of any level of dependence on the accuracy of ASTRAL, both when gene trees are known and estimated from sequences. We find that a fixed amount of dependence reduces the effective sample size by a constant factor. In current phylogenomic datasets, loci are generally sampled at large genomic distances to reduce gene tree dependence, thereby limiting the number of genes available for inference. However, full independence between genes is not required for accurate species tree estimation, and excluding gene trees may reduce inference accuracy. This creates a trade-off between the number of genes used and the degree of gene tree dependence. We therefore propose a method to identify the minimum genomic separation required to maintain satisfactory inference accuracy.