You can wear an armor piece for each limb plus your chest and head and then a thin garment underneath. This means you can wear your vault suit or some other clothing and have a piece of armor on each arm/leg and your chest/head, and these will be rendered on top of each other when you zoom out. 

Note that some items of armor will completely remove all other armor (for example the Haz Mat Suit will remove everything and you'll have to put it all back on again afterwards).