In reply to EmSB:
Any long term increase in the width of your fingers is likely to be very small - tendons don't gain bulk anywhere near as much as much muscles do.
However one likely cause for fatter fingers is short term swelling. After a heavy climbing session, all the tissues of the finger (including the skin) are a bit trashed, and hence can swell quite a bit. The effect is particularly noticeable in beginners, or people who come back to climbing after a long break. For regular climbers its still noticable, but less so.
I find that after a heavy session of bouldering my wedding ring is a bit trickier to push on, but the next day it's all back to normal. In contrast, when I started climbing again a few months ago (after a year's break) my hands were swollen for several days afterwards!