I would say it's definitely possible in Wellington but it entirely depends on your lifestyle. For our first two months here, we lived in a central apartment so we had no (free) parking and went without a car. Like Tigerlilly, we rented a car for weekends when we wanted, and walked pretty much everywhere.
At weekends we occasionally ventured out to the Wairarapa on the train, or on buses to check out suburbs we might want to live in. But overall, we missed the freedom and flexibility a car gives you.
We now live in a suburb that has great bus links into town, except we live at the top of a hill which is served by a bus route that is purely for commuters (weekedays and only runs into town in the morning and out of town in the evening). I couldn't dream of cycling right now, but we also have motorbikes as awell as a car, which is a much easier way to get around the city (and park!).
But ultimately, yes you can live in Wellington quite easily without a car. It just depends on where you live and how much freedom you want/need.