Place 2 cans of the black beans in a colander; rinse and drain.
Heat olive oil over medium-high heat. Add the ham and cook, stirring occasionally, until browned, 3-4 minutes. Add the onion, bell peppers, and garlic. Add in the Cumin and chili powder, and paprika. Cook, stirring occasionally, until fragrant, about 1 minute. Add the drained beans, broth, green chiles and remaining 2 cans of beans undrained; bring to a boil. Reduce the heat, and simmer until slightly thickened; about 30 minutes.
Remove the saucepan from the heat; allow to cool about 10 minutes. Pour 4 cups of the soup in a blender and puree. Return the pureed soup to the saucepan and stir in the salt. Cook, stirring occasionally, until heated through, about 2 minutes.