Researchers with the New England Centenarian Study at Boston University followed groups of people from their nineties to past age 110 to see how the timing of disease changed as people lived longer.
The study followed 104 people aged 110 to 119, 430 people aged 105 to 109, 884 centenarians, 343 people in their nineties, and 436 younger controls for an average of three years. The onset of cancer, heart disease, dementia and stroke was progressively delayed in each older age group, not avoided.
By the time people reached the supercentenarian range, near the outer edge of recorded lifespan, time spent sick before death had compressed into a shorter window, a pattern known as compression of morbidity. Living exceptionally long did not mean living disease-free, it meant disease arrived later. The cohort at the extreme end, 104 people past 110, remains small by epidemiological standards.

Read the original study
Andersen et al., Journals of Gerontology Series A, 2012 · doi.org/10.1093/gerona/glr223
Longer lives, later illness, not none.
Source
- Andersen, S. L., Sebastiani, P., Dworkis, D. A., Feldman, L., & Perls, T. T. (2012). Health span approximates life span among many supercentenarians: compression of morbidity at the approximate limit of life span. The Journals of Gerontology, Series A, 67(4), 395–405. https://doi.org/10.1093/gerona/glr223