What might a high hemoglobin indicate for the body?

High Hb. High hemoglobin may be normal in a person residing at high altitude. It may occur in patients with chronic lung disease, cyanotic heart disease, variants of hb gene that bind oxygen tightly, and dehydration. A type of blood cancer, polycythemia vera has high hb.
Sleep apnea. Obstructive sleep apnea is also in the differential diagnosis. For others, see dr. Singh's excellent answer for this question.