In syntactic dependency trees, when arcs are drawn from syntactic heads to dependents, they rarely cross. Constraints on these crossing dependencies are critical for determining the syntactic properties of human language, because they define the position of natural language in formal language hierarchies. We study whether the apparent constraints on crossing syntactic dependencies in natural language might be explained by constraints on dependency lengths (the linear distance between heads and dependents). We compare real dependency trees from treebanks of 52 languages against baselines of random trees which are matched with the real trees in terms of their dependency lengths. We find that these baseline trees have many more crossing dependencies than real trees, indicating that a constraint on dependency lengths alone cannot explain the empirical rarity of crossing dependencies. However, we find evidence that a combined constraint on dependency length and the rate of crossing dependencies might be able to explain two of the most-studied formal restrictions on dependency trees: gap degree and well-nestedness.