Remember that most of our TDS meters read in ppm. It only takes 50ppb of copper to nuke a tank. 50ppb (that's 0.05ppm) would read 000 on most TDS meters.
Yes, DI resin will pull copper out. It will pull it really well. And since there is so much more other stuff in your tap water, if you see 0 you know the DI is working so you can rest assured it's getting all the copper. Once it starts to creep up at all then all bets are off. Copper isn't the first thing that will break through, but it is pretty high on the list.