- It seems completely full stockpiles won't get a barrel/pot/bin hauled to it as a free space is needed for the container.
When there is an item blocking the stockpile tile, the container will still be assigned to the stockpile (you'll see the stockpile # added to the container's description, even if the container is far away), and items will be put in it, then it will be hauled to the stockpile and put in the same tile as the item.
If you didn't find a solution in what was already posted, to help you determine what is happening, you can check your 'j'obs menu. If you have any "store item in pot/barrel" jobs, hit 'j' again to see what items are being stored in that job.
It is like this... it is easy to have too much un-stockpiled food. You do some trading and butchering and suddenly have 400 food items that the dwarfs want to haul. There will be a rush of movement as many individual items are brought to the stockpiles, then all the movement will stop and you will wonder why they are not filling those barrels/pots. The problem is that each food stockpile will only have one "working" container, and only one dwarf will work it. He will do the work efficiently, but slowly, taking a couple of months to fill 30 containers. If this describes what you have been seeing, you can speed up the container-packing by creating more food stockpiles (for each stockpile, allow containers and only allow meat/fish/cheese/plant/leaves/cooked foods, the foods that can rot), then there will be more dwarfs working barrels and it will get done faster. Having four 20 tile stockpiles can be better than having one 100 tile stockpile.