I'm an intern in the Seattle area for the summer and was faced with a choice upon arriving - buy a bike and get $350 of it covered or pay ~$1000 to get a rental car for all three months. Due to the obvious financial benefit of the former I wanted a bike, but after it raining for a week straight upon my arrival I chose to get a car. This, along with the tremendous social benefits of being able to come and go as you please, convinced me that owning and riding a bike is a healthy alternate transportation method for the nice days, but not really a feasible replacement.
I have spent the last two years at university cycling to lecture and am now cycling to work. It saves a large amount of money (~£1800 insurance+£1500 running costs). If it rains you just wear waterproof clothing. You can easily get good quality waterproofs (e.g. Gore-Tex) for £200 which will last for years.
You have to be willing to live fairly near to work of course. For me it would probably take about the same amount of time to drive to work. It takes me about 13 minutes to cycle and rush hour traffic will slow a car down.
If I want to go longer distances I can catch a train (and take the bike for the other end) or take a bus. Occasionally hiring a car is also possible for rare occasions.
So for me using a bike is completely feasible for living in a city.
I'm from Vancouver Island(right near seattle) and bike through the winter. It really isn't that bad when it's raining, wear a jacket. The only days I don't bike are snow days(because they don't do snow removal very well here) and then I take the bus.
