SunnyBunny,
Read the terms and conditions which vary, slightly, from airline to airline, I suspect you will find that it is your responsibility to be at the gate on time.
You remember the terms and conditions don't you? That was the thing you agreed to when you had to check a box (saying you read and agreed with them) when buying on line.
It really depends on the airline if they will give you a refund or a credit. It wasn't their fault you didn't get there on time for your flight.