With co-ops, the key word is sharing. You are responsible for your own living space, but your building is responsible for some of it as well. The reason it gets complicated at all is knowing to what extent your building covers potential problems, and to what amount of your space do you need to have covered with your own co-op insurance policy.
In a condo or co-op you share the walls of your property, the hallways, stairs, elevators, and general building recreational and functional amenities. These will get covered by monthly maintenance fees and association dues. This master policy is only going to cover the common ground between you and your neighbors, not your belongings or the inside of your living space. You, and only you, are expected to maintain the quality and safety of things within your capability. That's where your personal co-op insurance comes into play.
You should always ask your building association certain things: 1. whether or not they offer "bare walls" coverage for structural and fixture damages; 2. if they have "single entity" coverage for things like carpeting, appliances, and ventilation inside units; and 3. if they offer "all-inclusive" coverage for all things in the condo or co-op minus your furnishings and personal belongings. Whatever they don't provide you must find in a policy of your own.
You should also be sure to ask if there's anything you need to be aware of insuring that was not mentioned previously. It's better to be safe than sorry.