Is it really impossible for Bumblebees to fly?

No it's an urban myth. It's impossible for them to fly using a very simple and inappropriate model of wing behaviour - possibly closer to say that bumble bees can't glide like albatrosses

Insect flight is different than bird flight. With insects, the rapidly moving wings, which do a figure 8 sort of motion, generates a vortex tube over the wings. This vortex by Bernoulli principle has less pressure, which permits the larger air pressure underneath to lift the animal up. If one is trying to understand insect flight according to the mechanics of bird or aircraft flight you are then pounding a square peg into a round hole, which makes you conclude bees can’t fly.

I would just like to add something here.

These answers are great but I might have another answer. At first it was indeed a mystery how these insects were capable of flying, but thanks to high speed recordings they found something which investigators didn't consider. The wing motion has a sort of double lift feature. By twisting her wings over at the end of each down stroke, the upward momentum is never lost. Basically, this means that even when the wings are going upwards, the wings provide upward lift.

(this also explains why, because bumblebees are so massive, they need a lot of muscle to get at the needed (Ca.) 300 flaps a minute)

I hope this was a complete answer! Cheers.