No.

1. NYC is _BIG_. It's 304 square miles, which is about 17 by 17 miles. There are many places where biking is too far.

2. Subways, in contrast to popular opinion, don't go everywhere. Mainly, unless you live in Manhattan, or that's the only place where you want to go, the Subway system is quite lousy. Going from the airport to Coney Island takes an hour and a half by subway (according to Google), rather than half an hour by car. Now what happens if you have what to do in Queens, Far Rockaway, or NJ?

3. Expensive and no point. Due to #2, most NYers have to have a car, if at least for those off days when they want to leave the city or what. Once you have the car, driving it is much cheaper than taxi.