Are nucleotides at the ends of DNA stripped on aging?
The Stanford article that you read is correct, in the sense that telomeres do not need to be completely removed by cell division before deleterious effects occur and cells start undergoing senescence. 

This Nature article describes an experiment in which the minimum length of telomeres (beyond which chromosomal fusion occurs) was determined:


  The critical telomere length was 12.8 repeats (of six base pairs)
  long, they found — any shorter and the chromosomes began to fuse
  together at their ends (R. Capper et al. Genes Dev.
  doi:10.1101/gad.439107; 2007). Baird suggests that without
  sufficiently long telomere caps, the cell may perceive the chromosome
  ends as broken strands of DNA and attempt to piece them together.


Therefore, we can conclude that while it is not impossible for non-telomeric nucleotides to be stripped from chromosomal ends, this generally does not happen, since cellular defense mechanisms such as non-homologous end joining (NHEJ) and microhomology-mediated end joining (MMEJ) would activate before that point, and start joining chromosomal ends together. This would then lead to issues during mitosis, which would likely halt further cell divisions.