Despite being such valuable ammunition for the technophobes and pseudoscientists of the world, there is actually no such thing as the non-flying bumblebee theorem. From a Google search, it’s clear that no-one really knows where the quote originated. My own theory is that it’s a mangled version of a subtly different statement: that “the equations of aerodynamics can’t be solved for the case of a flying bumblebee”. This latter statement is perfectly true – or at least it was when the myth of the non-flying bumblebee emerged in the 20th century.
The fundamental equations of aerodynamics are called the Navier-Stokes equations. These are non-linear partial differential equations, and they’re notoriously difficult to solve even with a modern computer. In fact they can’t be solved at all, in any general sense – they can only be approximated if you make additional simplifying assumptions relevant to the specific cases of interest. During the 20th century, the main “cases of interest” were manned aircraft. While still not easy, the Navier-Stokes equations could at least be made tractable with the aid of a number of simplifying assumptions – such as considering steady-state flow only, and restricting viscous effects to a thin boundary layer.
But these assumptions simply aren’t applicable to a bumblebee. The flow of air around such a tiny insect is highly viscous and anything but steady-state. If you tried to apply equations tailored for a 50-metre long, Mach 0.8 airliner to a bumblebee you might well calculate that the latter couldn’t fly – but the result would be meaningless, because the equations were applied way beyond their range of validity. 20th century aerodynamicists simply weren’t interested in developing the correct set of equations for bumblebees. The situation these days is different, with a lot of current research going into insect-sized “micro air vehicles”. Some of the devices emerging from science laboratories are described as bumblebee-sized – making it more ludicrous than ever to claim that scientists have "proved" that such things can’t fly!