Phylogenetic trees reconstructed from molecular sequences are often considered more reliable than those reconstructed from morphological characters, in part because convergent evolution, which confounds phylogenetic reconstruction, is widely believed to be rarer for molecular sequences than for morphologies. However, neither the validity of this belief nor its underlying cause is known. Comparing thousands of characters of each type that have been used for phylogenetic inference, we find that on average morphological characters indeed experience much more convergences than amino acid sites, but this disparity is explained by fewer states per character rather than an intrinsically higher susceptibility to convergence for morphologies than sequences. We show by computer simulation and actual data analysis that a simple method for identifying and removing convergence-prone characters improves phylogenetic accuracy, potentially enabling, when necessary, the inclusion of morphologies and hence fossils for reliable tree inference.