I had an interesting quick discussion on Twitter earlier today where the issue of the old "bumblebees can't fly myth" came up. This is an old story that has turned into something of an urban legend. There are all sorts of versions running around the internet (brief overview here), but they typically suggest that an aerodynamicist of some note showed that it was mathematically impossible for a bumblebee to fly. Obviously, bumblebees do fly, and the story is often used in a derogatory sense to put down scientists and engineers or being intelligent yet (in the views of the story-tellers) unable to grasp something obvious to everyone else. It is, as it were, another version of the Straw Vulcan argument to deride logical approaches to problems.
I could go on for a while about how ridiculous this sort of thing is, but I imagine it is self-evident to anyone that reads this blog, so I'll hold my typing on that front. Needless to say, the story is based upon a flawed understanding of the situation.
What was shown at some point (though the details are foggy when this was done the first time) is that a standard steady-state, oscillating foil model using a rigid wing does not accurately describe bumblebee flight. If bees had to fly like airplanes, they would indeed by grounded. But, of course, bees don't fly like airplanes. For one thing, insects have flexible wings. The shape changes in the wings are critically to their lift production. Second, insect wings carry a significant virtual mass of air on them as they move, owing to the "stickiness" of the air at that size scale. This means the effective profile of the wing is not the same as the anatomical profile. Thirdly, bees and many other insects with high wing beat frequencies seem to rely heavily on unsteady effects. This means that the air never reaches equilibrium on their wings. The flow is kept in a constant state of imbalance; this works best at very high wing beat frequencies and small size scales and can produce very high lift coefficients (well above steady state maxima). Small birds and bats can use some of these tricks, too, it would seem, because the measured maximum lift coefficients are often quite large in small animal flyers (about 5 for flycatchers, for example, compared to the 1.6 theoretical maximum for a thick feathered wing with slight camber).
So, in short: yes, bumblebees can fly. We understand how they fly and have for a long time, though there is plenty left to learn about insect flight overall. The story that someone proved bumblebees can't fly on paper is a myth (or, at best, a major misinterpretation). Finally, suggesting that logic is a bad way of solving problems is silly.