Do hemoglobin levels drop during menstruation?

Yes: Menstruation is blood loss, so anything but the lightest of cycles will cause the HGB to go down.

Yes with heavy bleed: Hemoglobin levels have been known to drop slightly around the time of periods if you have fairly heavy menstrual bleeding. If this continues, an ob/gyn evaluation for fibroids may be in order. Some people may have a very mild drop in hemoglobin (usually within normal limits) with fairly normal periods too, but the hemoglobin levels should not drop too low in this case.