In the study of the immune system, understanding the development of lymphocytes is crucial. Lymphocytes, which include B lymphocytes (B cells) and T lymphocytes (T cells), are essential components of the adaptive immune response. The term immunocompetent refers to the ability of these cells to effectively respond to pathogens or foreign substances.

Both B cells and T cells originate in the red bone marrow. However, their paths diverge after this initial stage. B cells remain in the red bone marrow to mature and become fully immunocompetent. In contrast, T cells leave the red bone marrow at an immature stage and migrate to the thymus, where they undergo further maturation and become immunocompetent.

Thus, the two key organs involved in the maturation of lymphocytes are the thymus and the red bone marrow. This knowledge is essential for understanding how the immune system prepares to defend the body against infections. In summary, the correct answer to the question regarding where lymphocytes become immunocompetent is both the thymus and the red bone marrow.