Is it possible to drive a car in chest high water?

Yes you can... provided that the engine's air intake doesn't get submerged in water. If water does get sucked into the engine, it will blow the seals and wreck the engine.

That's why you see these high mounted air intakes on serious offroad vehicles such as this:

http://www.jeep4x4center.com/cool-air-intake-with-...

It prevents water from getting sucking into the engine.

The other factor to consider is not submerging the alternator, battery and other electrical components unless they are also raised or shielded from the water.

Note that when going through water, there's a technique where you basically cause a wave in front of the vehicle and that prevents the electricals and other things from being as deep in the water.

In addition to what everyone else has said, you should not drive in water unless you know the surface under the water is safe. During a flood, it is quite possible that the moving water has washed away parts of the road surface, and you could drive into a hole or a ditch that you can't see, and you could end up with your car being washed off the road or under water completely.

as long as the air intake is above water and you have the correct electrics fitted, you can drive in deep water. However if the water is flowing fast there is a good chance the vehicle will be carried away by the current

I wouldnt recommend it but yes you should be able to if the exaust of the car is up at a fair hieght, if not then no you cannot drive in the water hieght.