Dave mentions his recent poor experience with American Airlines. I’ve had a number of those tight connections / run through the airport / arrive at gate / flight is closed / yet there is 5 or 10 minutes until departure time experiences.
However, I can recall two positive AA experiences:

• Extremely poor cabin conditions:
  

  I arrived at my middle seat (9E) for a redeye West Coast to Chicago flight only to find that my left seatmate (9D) was very, very big and had the armrest up while occupying some a portion of my seat. The woman to my right in 9F had a cat under her legs….

  I wrote a letter to AA’s CEO (Bob Crandall) and they promptly sent me my money back.

• Years later I flew through AA’s then hub in Nashville. My in bound flight was delayed and I faced a very tight connection to the last Dallas/Fort Worth plane of the evening. I stood up to depart my plane when my name was called. A bus drove me to the departing plane, just in time. That will probably never happen again.

I’ve also had many good experiences on United and Midwest Airlines. Your mileage may vary.