The hypothesis that all languages are equally complex often invokes a trade-off principle, according to which if a language is more complex in one particular domain, it will be simpler in another different domain. In this paper, we use data from WALS to test the existence of a trade-off between two specific domains: morphology and syntax. Contrary to widespread views, we did not find a negative correlation between these two language domains, but in fact a positive correlation. At the same time, this positive correlation seems to be driven by some language families, and it disappears when one considers purely morphological and purely syntactic features only. We discuss these findings in relation to ongoing research about language complexity, and in particular, the effects of factors external to language on linguistic structure.