Some very good points for the existence of an upper limit to how old humans can get are given in this paper by Dong, Milholland & Vijg in Nature (but see critique of stats and statements, linked below).
Vaupel and colleagues had argued previously that there was no such limit, based on analyses of Swedish cohorts that showed a steady increase of of maximum lifespan over the last 100 years up to the 1990s. In their paper, Dong et al. argue that this increase peaked in the mid 1990s, coincidentally that time when the world record holder of oldest age, Jeanne Calment, died at 122 years of age.
The authors close with these statements:
“To further extend human lifespan beyond the limits set by these longevity-assurance systems would require interventions beyond improving health span, some of which are currently under investigation. Although there is no scientific reason why such efforts could not be successful, the possibility is essentially constrained by the myriad of genetic variants that collectively determine species-specific lifespan.”
First reactions can be found here: https://www.nrc.nl/nieuws/2016/10/07/human-lifespan-limited-to-115-years-a1525476
Fun links (and data that was actually used in this paper):
The Human Mortality Database: http://www.mortality.org/
Supercentenarian (>110 years of age) List of the Gerontology Research Group: http://www.grg.org/SC/WorldSCRankingsList.html