We analyze incidence and evolution of patent thickets. The paper provides a modeling framework showing how competition for patent portfolios, complementarity of patented technologies and hold-up affect patenting. Predictions are that technological opportunity reduces patenting in complex technologies, while increasing patenting in discrete technologies. Competition has the opposite effects. The predictions are tested using European patent data in a panel with 2,074 patenting firms in thirty technology areas over fifteen years. A new measure of technological complexity is applied. GMM estimation results confirm the predictions of our preferred model. Patent thickets are found in nine out of thirty technology areas.