Nemerteans (phylum Nemertea) have been viewed by mostzoologists as descended from, or closely related to,the flatworms. This view is based mainly on theirsupposedly acoelomate body. Their ancestry, however,is a point of controversy and there is evidence for acoelomate, protostomous origin. Notwithstanding thesedifferent views, most zoologists consider nemerteansto be phylogenetically distant from the chordates.Four authors (Hubrecht, Macfarlane, Jensen, Willmer),however, have postulated that nemerteans instead areclosely related to the chordates and that they sharea most recent common ancestor with the vertebrates. We argue that this view is based on a flawed view ofhomology and of seeing evolution as a series ofprogressions, which has no support in modernevolutionary thinking. Since there are nomorphological synapomorphies supporting aChordata-Nemertea clade, these authors instead guesswhat characteristics in extant nemerteans gave rise tocharacters observed in recent chordates. For example,they propose that the nemertean proboscis sheath hasevolved into the notochord. This is mere speculation,lacking testable propositions and is hence void ofinformation, and thus becomes futile in our view. However, the idea of a nemertean-vertebrate sisterrelationship as such is a testable hypotheses, and wetest it by applying the parsimony criterion to a setof morphological characters, and a set of molecular(the 18S rRNA gene) characters. Both tests reject thehypothesis.