Bott-Tu has a comment in the introduction about Poincaré not discovering the computability of de Rham cohomology through combinatorial data associated to a finite good cover. I might as well give you the original:

"To digress for a moment, it is difficult not to speculate what kept poincaré from discovering this argument forty years earlier. One has the feeling that he already knew every step along the way. After all, the homotopy invariance of the de Rham theory for $\mathbb{R}^n$ is known as the Poincaré lemma! Nevertheless, he veered sharply from this point of view, thinking predominantly in terms of triangulations, and so he in fact was never able to prove either the computability of de Rham or the invariance of the combinatorial definition. Quite possibly the explanation is that the whole $C^\infty$ point of view and, in particular, the partitions of unity were alien to him and his contemporaries, steeped as they were in real or complex analytic questions."

EDIT: I guess this is more of a failure to observe the fact, rather than it being opposite to expectation. Nevertheless, it's interesting and perhaps close enough.