Common cancer-related causes of low blood cell counts include:

Chemotherapy. Certain chemotherapy drugs can damage your bone marrow — the spongy material found in your bones. Your bone marrow makes blood cells, which grow rapidly, making them very sensitive to the effects of chemotherapy. Chemotherapy kills many of the cells in your bone marrow, but the cells recover with time. Your doctor can tell you whether your specific chemotherapy treatment and dose will put you at risk of low blood cell counts.
Radiation therapy. If you receive radiation therapy to large areas of your body and especially to the large bones that contain the most bone marrow, such as your pelvis, legs and torso, you might experience low levels of red and white blood cells.
Cancers of the blood and bone marrow. Blood and bone marrow cancers, such as leukemia, grow in the bone marrow and don't allow normal blood cells to develop.
Cancers that spread (metastasize). Cancer cells that break off from a tumor can spread to other parts of your body, including your bone marrow. The cancerous cells can displace other cells in your bone marrow, making it difficult for your bone marrow to produce the blood cells your body needs.