I am a British Asian who has been asked the question in the title for a few hundreds for the past few years. In the past, I didn’t take issue with it, but it is getting more and more offensive as many folks seem never to be contented with any answers you give them on the basis of your surname or outlook. They simply assume you can’t be British when you are not White (dominant racial group in Britain). This is the worst when it comes to looking for a house, many landlords / flatmate-searchers ask these loaded questions after you tell them where you are living in as they are simply incredulous about your connection to the country. It’s very disheartening. How many decades more I have to live in Britain in order to be considered British?