That is the thing. The latter is impossible so long as we're fleshy meat bags. The former, some kind of treatment or pill that stops or reverses aging might be.
I'd have to search for it again but there was a statistical analysis that factored in accidents (plane, train, automobile, falling off a ladder, slipping in the shower, etc.), weather events and natural disasters, and medical incidents. There was a rough upper limit of 2000 years before it was practically guaranteed the odds would catch up to you. Immortality isn't all that long really when you look at it.
The thing is that age itself isn't what kills you. It's the susceptibility to physical dangers or medical issues that does. You won't find "old age" on a death certificate. You'll find cancer, renal failure, stroke, trauma from falls, etc.