Absolutely, an anorexic could not know they have anorexia. That's part of the problem, and how it can gain such a foot hold in person's mind. It's an extremely serious illness and requires immediate treatment. The recovery process can be slow and there can be setbacks, but it's definitely possible to overcome it.
Not all anorexics know they have the condition. The condition is known to be a serious mental illness that makes the person not eat or eat very small amounts of food because they have an obsessive fear of gaining weight. Among psychiatric disorders, it has one of the highest death rates. Part of treatment means psychological therapy. More information can be found at http://en.wikipedia.org/wiki/Anorexia_nervosa.