It suddenly struck me that the English word beanstalk and the German word Bienenstock (beehive) sound powerfully like cognates, even though they are not. There are quite a lot of faux amis between English and German, and they are usually cognate, even when the meanings are radically different — as between the English fabric and the German Fabrik (factory), or the English stuff and the German Stoff (fabric). They have a common root, from which they have evolved differently. Even the bizarre Gift meaning “poison” started out as something given, a dose of medicine (dosis also from the Latin root for “given”).
But beanstalk and Bienenstock are both compound words made up of parts that both seem like they could be cognates, but actually are unrelated. That beans and bees are unrelated is unsurprising. It took me a bit of work to convince myself that stalk is etymologically unrelated to Stock, which is indeed cognate to the English stick. The roots are quite different: Stalk from Old English stale, meaning a handle or part of a ladder; Stock originally a branch or a treestump, presumably then a stump that houses bees, either naturally or agriculturally.