My wife and I tried this a few years ago for a few weeks, either bringing in our own Tupperware or Mason jars to do it, and each time the person working in the deli area had to get a manager involved, and one time they accused us of stealing their tupperware, since they sold that same container in the shop. We did this partly for environmental reasons, but mostly for "meat packaging gets really stinky in your garbage can after a few days in a small apartment" reasons. After awhile, we decided it wasn't worth the headache to us.

In NYC (and California too I think?), they're starting to encourage bringing your own bag instead of buying them at the register, so I might retry this experiment again.