It is possible to develop lactose intolerance but not necessarily normal. Every human to a degree is lactose intolerant. Allergies change periodically and many people who don't seem to have an issue with milk can have an issue at any given time.
Yes, in fact this is rather common that people become lactose intolerant as they get older. Depending on severity, you may still be able to eat yogurt, cheese, and things where the milk is cooked (creamed soups, custards) with no problem.