Of course it's possible to have six-wheel drive, there are military vehicles which are ten-wheel drive.
All starts to get rather heavy and complex after the second axle though.
There have been two 'styles' to 6 wheeled formula 1 cars. The Tyrrells with their four front wheels, and the Marches with their four rear wheels. The object behind the Tyrrel was that you have double the grip on the front since you have double the amount of tyre in contact with the ground. It worked, and basically kicked butt in any GP it ran in, until the F1 crowd decided to change the rules to ban them.
The idea behind the four rear wheels on the March six-wheeler was slightly different. Back in those days, the difference in size between the front and rear wheels was much more pronounced than today, which is saying something. They decided that if you had four smaller wheels at the back, you would get as much, if not more grip to put the power onto the track, and at the same time drastically reducing the frontal cross-section, and thus drag. It never raced due to the changing of the rules.
F1 is known to change rules midstream to ban innovative though. Take, for example, the car that used basically a vacuum system to 'hug' the track. It raced once, and won, then got banned.
And just to show not just the Europeans were at it, here's one from Indianapolis, 1957.
NTM