Detection and attribution of anthropogenic climate change signals in sea level rise (SLR) has experienced considerable attention during the last decades. Here we provide evidence that superimposed on any possible anthropogenic trend there is a significant amount of natural decadal and multidecadal variability. Using a set of 60 centennial tide gauge records and an ocean reanalysis, we find that sea levels exhibit long-term correlations on time scales up to several decades that are independent of any systematic rise. A large fraction of this long-term variability is related to the steric component of sea level, but we also find long-term correlations in current estimates of mass loss from glaciers and ice caps. These findings suggest that (i) recent attempts to detect a significant acceleration in regional SLR might underestimate the impact of natural variability and (ii) any future regional SLR threshold might be exceeded earlier/later than from anthropogenic change alone.