It can happen with certain Non-accredited Allo fellowships. Problem is, given our countries history of slavery and indentured servitude we are extremely wary of arrangements where people are being asked to work long hours for no pay, no benefits, etc. And a program gives up a lot of control over it's resident/fellows if they aren't drawing a salary that can be docked. The threat of firing someone becomes meaningless if it wasn't a real paid job to start with. There's only so much influence you have over someone who is working for you just to say they did, as a means to an end. You get what you pay for. And most places feel the need to line up moonlighting opportunities for unpaid people, so that creates a loss of controllable time and duty hours and adds additional administrative burdens. All if which is why at the resident level, no pay isn't a real option.
