This happened to me once, travelling on Delta. A 1:15 connection (in Atlanta) became a 47 minute connection. It was a legal connection time, but one that I wasnt comfortable with. I'm a flight attendant, and I see what can happen to people with short connections.
In this case, I knew that even a reasonable 20 minute delay for de-icing at my departure airport, plus the time it takes to get off the plane (and wait for gate checked bags in some cases) would make the connection very stressful. I also knew that even though there were a couple more flights to Orlando that night, they'd likely be full by the time my travel date came around. They can't put you on a flight if there are no seats, and if the missed connection is due to weather delays, the airlines have no obligation to pay for hotels while you wait for a flight that does have seats.
I simply called the airline, told them that I was uncomfortable with the shorter connection time, and they offered to change me to the next flight, no extra charge.
It was a good call: a snow storm hit just as we were boarding, and we were delayed 25 minutes in the deicing bay. I still tried to make the earlier connection to see if I could get on. I got to the gate just as the plane was pushing back. I had a chat with the gate agent afterwards, who confirmed that the rest of the flights to Orlando were oversold for the next 3 days. I would have lost at least a full day (or more) at Disney, would have had to find and pay for my own hotel, plus the loss of non-refundable Disney hotel nights.
My advice: Better to be safe than sorry; request to be changed to the next connecting flight, then take the chance on being able to standby early.