Back in May I was driving from Central Wisconsin to Madison for a flight. My routing was MSN - ORD - PHX. About 10 miles from the airport, while still on the highway I received an alert that my flight was delayed 30 minutes. This put me at a tight connection (30-40 minutes if I recall). I decided to keep driving down to ORD - it's only a 2 hour drive and I would be able to arrive at ORD with 2 hours to spare. I immediately picked up the phone and called and explained that my MSN to ORD flight was delayed and I wanted to forgo that segment and just take the ORD to PHX flight. The agent did push back a bit as it wasn't a huge delay but I explained that I've done the flight quite often and I know that a 30 minute delay typically gets pushed back and I'd like to play it safe (not to mention you large roller boards can't go in the overhead). The agent did agree to make the change but stated there would be no compensation or refund for that segment which I agreed to. It did take the agent some time to get everything in order and I had to check in again for my ORD to PHX flight.
I've also done this the other way when going ORD to MSN. Those short flights are always quick to get cancelled, especially with weather.
So, it can be done but anytime you do this you HAVE to call. You can't just show up for your next flight and expect to board with no issue. I'd also say you need to be clear that you are calling to make the change rather than just showing up as OP seems to imply.
If you get an agent that says no I suggest you HUACA.