The other day I was one of the last passengers to board a (domestic US) flight, and my carry-on bag was checked at the gate without asking. I suppose this was due to lack of space in overhead bins (however there was room right above my seat). What if my carry-on consisted entirely of fragile items? For example some kind of neatly packed set of wine glasses or something similar? Could I refuse to check the bag?
The only space you have a right to is the space underneath your seat. And even then, some aircraft entertainment systems take up much of that legroom.
So if your bag won’t fit underneath your seat and you refuse to board without your bag with you, they may bump you to the next flight and let you board that one earlier.
If there is no next flight or the flights are all on smaller planes, they could just refuse to board you unless you check the luggage and sign a damage waiver. In that case, your recourse is limited. You could use a delivery service instead to mail your fragile items.