Generally, yes.

However, at the end of last year, I was flying back home (origin-DFW-SFO). DFW-SFO had a two-hour delay posted, and as I was settling in to work, there was a rushed announcement that "The doors are closing for Flight XYZ to San Francisco, please board immediately." About 15 people ran to the gate to find out the flight had almost pushed back without all the AC members. As a reward for our loyalty (and AC membership dues), the GA was collecting luggage for gate-check because everybody else was on the plane.

Only time that I have ever seen a delayed flight board/depart early.