Formal constraints on crossing dependencies have played a large role in research on the formal complexity of natural language grammars and parsing. Here we ask whether the apparent evidence for constraints on crossing dependencies in treebanks might arise because of independent constraints on trees, such as low arity and dependency length minimization. We address this question using two sets of experiments. In Experiment 1, we compare the distribution of formal properties of crossing dependencies, such as gap degree, between real trees and baseline trees matched for rate of crossing dependencies and various other properties. In Experiment 2, we model whether two dependencies cross, given certain psycholinguistic properties of the dependencies. We find surprisingly weak evidence for constraints originating from the mild context-sensitivity literature (gap degree and well-nestedness) beyond what can be explained by constraints on rate of crossing dependencies, topological properties of the trees, and dependency length. However, measures that have emerged from the parsing literature (e.g., edge degree, end-point crossings, and heads’ depth difference) differ strongly between real and random trees. Modeling results show that cognitive metrics relating to information locality and working-memory limitations affect whether two dependencies cross or not, but they do not fully explain the distribution of crossing dependencies in natural languages. Together these results suggest that crossing constraints are better characterized by processing pressures than by mildly context-sensitive constraints.