Most renovation and cleanout debris loads without any issue. The limits come from Connecticut's recycling law and from what the City's transfer station turns away.
Connecticut bans several things from the trash outright. Televisions, monitors, printers, and computers have to go to an electronics program, and so do car and lead acid batteries. Grass clippings are banned as well, and leaves have to be composted. Paint, mattresses, tires, and gas cylinders are handled through their own statewide take-back programs.
Hazardous waste, liquids, paint, solvents, fuel, and asbestos can't go in a standard dumpster either. Stamford's transfer station specifically refuses chemicals, hazardous waste, liquids, and oil-based paints and thinners.
Tires and appliances with refrigerant are accepted at the City scale but billed per item. Ask your account manager about anything you're unsure of before it goes in the container.