Darvin: This is one of the most irritating things about health care in Frostpunk: just because the user interface indicates that all your people are currently in care, doesn't mean that they are. A person counts as being "in care" so long as there is sufficient space at a health care facility for them, but they can still die before actually reaching that facility. But it gets worse! Engineers at medical posts can go on break to rest or eat (or become sick themselves), and if all five engineers at the medical post are off-duty then the post shuts down. Provided there's enough space at another medical post you won't get an alert that the patients are no longer in care as they walk from one post to another.

Gravely ill have only a certain amount of time to reach health care before they die, but this death timer does not reset upon reaching health care, so every time they are disrupted and need to walk to a new facility in this fashion the death timer counts down a little more. On the higher difficulties (hard and extreme) the death countdown is so unforgiving that sustain life is pretty much a death sentence to your gravely ill, and the law simply does not function.

TL;DR: take radical procedure to cure your gravely ill, it's the only way to keep them alive if you lack infirmaries or houses of healing.