Is it possible for somebody to be both anorexic and bulimic?

Yes: Yes -- some people with anorexia not only severely control the amount of food they eat, but also purge through vomiting, using laxatives & diuretics, etc. They may also exercise way out of proportion to the amount needed for enjoyment or physical health. They don't usually binge on large quantities of food before making themselves vomit, however.