In a skillet, cook bacon over medium-high heat until crisp, about 7-8 minutes. Transfer the bacon to a slow cooker with a slotted spoon, reserving the drippings in the pan.
Add the onion to the bacon drippings in the pot and cook, stirring occasionally, until soft, about 5-7 minutes. Add the garlic and cook for stir for 1 more minute. Pour 1 cup of the broth into the skillet, stirring with a wooden spoon to scrape up any browned bits from the bottom of the pan.
Transfer the broth, onion, and garlic mixture to the slow cooker. Pour in the additional 2 cups of broth, condensed soups, corn, broccoli, cream-style corn, and diced tomatoes. Stir to combine.
Cover the slow cooker and cook on LOW for 4-6 hours or on HIGH for 2-3 hours, until the soup is warm and the broccoli is tender.
Add Velveeta, cheddar, and chicken 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 salt and pepper, if necessary.
Ladle into bowls and enjoy warm!