I lived in Austin from 2007-2009, and while I did not bike, I used the public transportation extensively.

My take is that you could go car free in Austin if you lived within a few miles of downtown. While the summers are indeed opressive and often horrible, it seldom gets below freezing and you will be able to commute year around without having to worry about ice and snow (although a heat stroke is a possiblity...).

I used my car only a couple of times a week in Austin, mainly for getting groceries and running other errands. I would strongly recommend a car for the following reason:

Austin is a small city. You'll be confined to the central portions of a metro area with less than 2 million people. Want to go out to Lake Travis? You'll have a fairly long ride on curvy roads with much more traffic than they were designed for. A weekend in San Antonio or Dallas? I suppose you could fly it, but neither of those cities have great public transporation and you'd need rent a car when you got there. Travel to South Padre for a Texas beach vacation? Have fun riding the 300 miles. Austin is a smaller city in a BIG state, unless you are OK with central-ish Austin 100% of the time, get a car.

However, I think Austin has a car sharing program now...I don't know the specifics as it started after I left, but it might make living carless easier there.