Is it possible for one person to have several types of headaches?

Yes. It is possible to have more than one type of headache. Consult a pain physician or a neurologist for a comprehensive evaluation and management plan; it can get frustrating to have frequent headaches.
Yes. Assuming the headache is not caused by something more serious like stroke, increased intracranial pressure, or an inflammatory disorder like giant cell arteritis, one can have a combination of the 3 most common types of headaches: cluster, tension, and migraine.