Humans have genetically based unique abilities making complex culture possible; an assemblage of traits which we term ``cultural capacity''. The age of this capacity has for long been subject to controversy. We apply phylogenetic principles to date this capacity, integrating evidence from archaeology, genetics, paleoanthropology, and linguistics. We show that cultural capacity is older than the first split in the modern human lineage, and at least 170,000 years old, based on data on hyoid bone morphology, FOXP2 alleles, agreement between genetic and language trees, fire use, burials, and the early appearance of tools comparable to those of modern hunter-gatherers. We cannot exclude that Neanderthals had cultural capacity some 500,000 years ago. A capacity for complex culture, therefore, must have existed before complex culture itself. It may even originated long before. This seeming paradox is resolved by theoretical models suggesting that cultural evolution is exceedingly slow in its initial stages.