Thanks to all for your replies. The pump seems to be fine - works fine on the front tire.
I think it is the tube - I removed the tube, inflated it a bit, and it deflated pretty quickly. Did a quick check and didn't see a hole. I should note it is an unused tube, but it had been sitting in my saddle bag for about eight years, so maybe it just exceeded its shelf life?
In any event, I'll head to the LBS and get a couple new tubes and try again. Thanks again!