Airlines pretty well have to accept them as what else are they going to do. Make the passenger abandon the bag if it won't fit on board, or abandon the passenger with his bag? The more relevant question is what sort of fine the airline chooses to impose on the passenger for attempting to bring on board a bag that won't fit. In the US, by and large, that's not an issue, but elsewhere it will often be a larger amount than checking the bag in the first place.