Preheat the oven to 350 degrees F (175 degrees C).
Place chicken breasts into a large pot and cover with water; bring to a boil. Reduce heat to medium-low. Simmer until tender, about 20 minutes; drain. Shred chicken.
Combine sour cream, cream of chicken soup, and green chiles in a bowl. Spread a small amount of the mixture over the bottom of a 9x13-inch baking pan. Sprinkle 1/2 the shredded chicken on top. Sprinkle 1/2 the chips over chicken layer and top with 1/2 the Cheddar cheese. Repeat layering once more with remaining sour cream mixture, chicken, chips, and Cheddar cheese. Cover with aluminum foil.
Bake in the preheated oven until top is golden brown and bubbling, about 30 minutes.