Over the past 15 years, there has been increasing use of linguistically annotated sentence collections, such as the Penn Treebank (PTB), for constructing statistically based parsers.While these parsers have generally been built for engineering purposes, more recently such approaches have been advanced as potentially cognitively relevant, e.g., for addressing the problem of human language acquisition.Here we examine this possibility critically: we assess how well these Treebank parsers actually approach human/child language competence.We find that such systems fail to replicate many, perhaps most, empirically attested grammaticality judgments; seem overly sensitive, rather than robust, to training data idiosyncrasies; and easily acquire "unnatural" syntactic constructions, those never attested in any human language.Overall, we conclude that existing statistically based treebank parsers fail to incorporate much "knowledge of language" in these three senses.