One of the major efforts in molecular biology is the computation of phylogenies for species sets. A longstanding open problem in this area is called the Perfect Phylogeny problem. For almost two decades the complexity of this problem remained open, with progress limited to polynomial time algorithms for a few special cases, and many relaxations of the problem shown to be NP-Complete. From an applications point of view, the problem is of interest both in its general form, where the number of characters may vary, and in its fixed-parameter form. The Perfect Phylogeny problem has been shown to be equivalent to the problem of triangulating colored graphs. It has also been shown recently that for a given fixed number of characters the yes-instances have bounded treewidth, opening the possibility of applying methodologies for bounded treewidth to the fixed-parameter form of the problem. We show that the Perfect Phylogeny problem is difficult in two different ways. We show that the general problem is NP-Complete, and we show that the various finite-state approaches for bounded treewidth cannot be applied to the fixed-parameter forms of the problem.