Paranoia

I have setup an area that is just sets of diamond chests, stacked three high, and 15 blocks long. I have connected a teleport pipe, and I have redstone pipe going to each chest. I would like to set something up so that items that come in are evenly distributed among the chests. Right now it seems to just dump everything in the first chest only. Do I need to worry about items overflowing out of the chest and onto the floor when the first chest fills up? It is setup like this (+ is redstone pipe,T is teleport pipe, 0 is a chest, top down view): 000000000000000 +++++++++++++++++ 000000000000000 +T 000000000000000 + +++++++++++++++++ 000000000000000