Many fellowships can actually coordinate to run half-time split between two fellows where they work their day job half the time and the fellowship the other half. I know our Addiction fellowship was coordinating that for 2 prospective fellows.
I have never really seen this happen for residency, but I've heard that a special situation involving a couple splitting IM residency due to the family situation has happened. The logistics of that seems quite complicated though, and I can't imagine it's common.
To be honest, you graduated long enough ago, that this in particular is just not the case anymore. It's funny talking to attendings that were residents when I was an intern that are surprised by this, but the presence of NPs/PAs on services and throughout the residency education system has significantly increased over just the last 3-4 yrs. Half the time you're consulting an NP or PA here. This has happened in other academic institutions as I've heard as well. It's partly why when the FM RRC recommended changes to allow non-physicians as core faculty and supervisors, it really wasn't a surprise.
As an example of how this ends up directly damaging our training, they've expanded our PA "residency", and these PAs weren't even in PA school before we became residents. They sometimes take priority on procedures, despite them having better hours, better retirement benefits, and better pay than us, and we're told, they "only have a year" to learn this stuff or "have to go home by 5" as justification. We brought this up recently with literally the APD and they were shocked, despite them graduating barely 6 yrs ago.
Everyone who is wondering why current residents and med students are so bitter at all the NPs and PAs, it's because we've watched our academic centers favor them over us or our education. We've watched some of our attendings constantly give them a greater benefit of the doubt than us. All at the same time having to deal with the fallout of their treatment decisions. And this is coming from someone who actually believes they can be valuable, but not as independent medical practitioners.