If the battery is dead flat, you're pooched, but you can get way farther than you have any right to push starting a car with an only mostly dead battery because you don't have the (huge) load of the starter motor bringing the voltage down. I've had a couple alternators go and push started the cars they were in until I was able to replace the alternator.

On the basis of this experience, I'm not convinced the alternator actually comes into play in a typical push start. It's usually roll the car, clutch out, lurch and fire, clutch back in and let the engine get to a stable idle. At no point is the engine spinning fast enough to create much electricity with the alternator until after it's actually running. Provided the alternator is working in the first place, of course.

As an aside, in all of the vehicles where I've lost the alternator, the first warning sign has been the radio having a shit fit. I have never once seen the idiot light come on for a bad alternator, which really calls its utility into question.