Compound climate extremes are receiving increasing attention because of their disproportionate impacts on humans and ecosystems. However, risks assessments generally focus on univariate statistics. We analyze the co-occurrence of hot and dry summers and show that these are correlated, inducing a much higher frequency of concurrent hot and dry summers than what would be assumed from the independent combination of the univariate statistics. Our results demonstrate how the dependence structure between variables affects the occurrence frequency of multivariate extremes. Assessments based on univariate statistics can thus strongly underestimate risks associated with given extremes, if impacts depend on multiple (dependent) variables. We conclude that a multivariate perspective is necessary to appropriately assess changes in climate extremes and their impacts and to design adaptation strategies.