Preheat oven to 350 degrees F (175 degrees C).
Bring a pot of water to a boil. Cook chicken in boiling water until no longer pink in the center and the juices run clear, 7 to 10 minutes. Remove chicken to a cutting board to cool, retaining at least 6 cups water in the pot. Cook spaghetti in the boiling water, stirring occasionally, until cooked through but firm to the bite, about 12 minutes; drain. Melt butter in a large pot over medium heat. Cook and stir onion and bell peppers in hot butter until softened, 5 to 7 minutes; add Nacho’s cheese and cook, stirring occasionally, until the cheese is melted and smooth. Cut chicken into small pieces and stir into the cheese mixture. Stir drained spaghetti, peas, mushrooms, sliced olives, and chopped olives into the cheese sauce; divide between prepared baking dishes.
Bake in preheated oven until bubbly, about 45 minutes