Not impossible. I was zipping down a country road once, much faster than I should have been, and a large flock of sparrows made a tactical error and swooped right down in front of me. I plowed right through them and still remember the tiny "pop pop pop" sound of all the ones I wiped out in the encounter. I could see them falling to the ground in my read view mirror. I must have taken out at least a dozen with my little car. It was sad.
more coffee...
I had a similar situation happen to me on the Colonial Parkway near Williamsburg, VA. It was only a small handful of them, but for some reason they decided to light on the road when my ETA was .5 seconds. Way too late to maneuver to avoid them -- I tried to hit the brakes but ended up taking out the whole lot of them anyway.
But hundreds, sometimes thousands of birds inexplicably dropping from the sky is nothing short of weird.