Yes. I knew someone who lost a lot of blood once his skin was almost pure white. The hospital held off on doing a transfusion as long as they could because it is better if the body regenerates blood on its own. He did end up needing a transfusion tho.
Unless I am misunderstanding your question, the human body does produce blood by itself. It is produced in the bone marrow. During childhood almost every bone in the body manufactures blood. In adults the process is performed by larger bones
It can and does every single day.