As the application of genomic data in phylogenetics has become routine, a number of cases have arisen where alternative datasets strongly support conflicting conclusions. This sensitivity to analytical decisions has prevented firm resolution of some of the most recalcitrant nodes in the tree of life. To better understand the causes and nature of this sensitivity, we analyzed several phylogenomic datasets using an alternative measure of topological support (the Bayes factor) that both demonstrates and averts several limitations of more frequently employed support measures (such as Markov chain Monte Carlo estimates of posterior probabilities). Bayes factors reveal important, previously hidden, differences across six “phylogenomic” datasets collected to resolve the phylogenetic placement of turtles within Amniota. These datasets vary substantially in their support for well-established amniote relationships, particularly in the proportion of genes that contain extreme amounts of information as well as the proportion that strongly reject these uncontroversial relationships. All six datasets contain little information to resolve the phylogenetic placement of turtles relative to other amniotes. Bayes factors also reveal that a very small number of extremely influential genes (less than one percent of genes in a dataset) can fundamentally change significant phylogenetic conclusions. In one example, these genes are shown to contain previously unrecognized paralogs. This study demonstrates both that the resolution of difficult phylogenomic problems remains sensitive to seemingly minor analysis details, and that Bayes factors are a valuable tool for identifying and solving these challenges.