Well, not with traditional feedforward networks (LeNet, etc.). You can't run the classifier and find tires, then wheels, and then a car; but you do get composition of features.
I'd argue they are implicitly doing this.
> You can't run the classifier and find tires, then wheels, and then a car;
Why can't you run a classifier for tires, one for wheels, one for cars, then combine their outputs for a final classifier maybe based on a decision tree? You can train all the networks at the same time and it will give you a probability distributions for all 4 outputs (tires, wheels, cars, blended). What am I missing?
You're going to need large amounts of fine-grained labeled data for each category. You've also just manually determined some sort of (brittle) object ontology. What if there are only 3 tires? What if there are four tires on the road but no car? All sorts of edge cases, and all you've done is train a classifier for cars, not actually solved driving in any meaningful way.