Place broccoli, onion, potato, carrots, chicken broth, water, cream of celery soup, salt, and pepper into a slow cooker. Cover and cook on LOW for about 6-8 hours or on HIGH for about 3 hours.
Use an immersion blender to puree a portion of the soup. You still want plenty of chunks of potatoes and veggies, so only puree some of it. If you don’t have an immersion blender, you can put a cup or two of soup in a regular blender to puree (and then return back to the slow cooker when done).
Add Velveeta, cheddar, and ham to the slow cooker. Stir, and then place the lid back on the slow cooker and cook for another 15-20 minutes. Stir it again to make sure that the cheese is completely melted and smooth.
Give it taste, and season with additional salt and pepper, as necessary.
Ladle into bowls and enjoy warm!