A forum FAQ deals with what is a tight flight connection time. I had a scheduled 58 minute connection yesterday at SEA-TEC. My flight from MSP was delayed by 63 minutes (mechanical) but the pilot made up about a half hour arriving at the SEA-TEC gate only 34 minutes late. That gave me 24 minutes to make the last flight of the day to my home city. Of course it take about 10 or so of those 24 minutes to open the door and have 18 rows of passengers closer to the door exit. The crew suggesting that passengers not making tight connections let us through letting me get a couple of rows forward before passengers starting unloading the overheads.
Of course the MSP flight arrives at the end of Terminal D and my connecting flight is at the end of Terminal C and a regional turboprop at ground level without a jetway. Having a RS Classic backpack and holding my cross-body strapped messenger bag I could sort of run/jog. I made my connecting flight with about 5 minutes to spare and wasn't the last to board.
Interesting side note is both of my checked bags, including my oversized ski box, made the connection and were out for pickup less 15 minutes after the aircraft's doors opened.
Lesson is that a hour connection at SEA-TAC with a hour's departure delay is possible with a tail wind and running. And your checked bags can make the connection too.