Alignments of nucleotide or amino acid sequences may contain a variety of different signals, one of which is the historical signal that we often try to recover by phylogenetic analysis. Other signals, such as those arising due to compositional heterogeneities, among-lineage and among-site rate heterogeneities, invariant sites, and covariotides, may interfere adversely with the recovery of the historical signal. The effect of the interaction of these signals on phylogenetic inference is not well understood and may, in many cases, even be underappreciated. In this study, we investigate this matter and present results based on Monte Carlo simulations. We explored the success of four phylogenetic methods in recovering the true tree from data that had evolved under conditions where the equilibrium base frequencies and substitution rates were allowed to vary among lineages. Seven scenarios with increasingly complex conditions were investigated. All of the methods tested, with the exception of neighbor-joining using LogDet distances, were sensitive to compositional convergence in nonsister lineages. Maximum parsimony was also susceptible to attraction between long edges. In many cases, however, phylogenetic inference methods can still recover the true tree when misleading signals are present, in some instances even when the historical signal is no longer dominant. These results highlight the growing need for simple methods to detect violation of the phylogenetic assumptions.