WASHINGTON — Debate on the health care bill was coming to a close on a cold day in December when a senator speaking on the floor overshot his allotted time.

The man occupying the chair's seat that day was a Democratic freshman, who had arrived in Washington some months earlier after besting a Republican incumbent in a tough, bitter race. The health care bill was in a race against the clock; required cloture motions needed to be squeezed in so a vote could take place in time for lawmakers to...