Telomeres protect the ends of linear genomes, and the gradual loss of telomeres is associated with cellular ageing. Telomere protection involves the insertion of the 3' overhang facilitated by telomere repeat-binding factor 2 (TRF2) into telomeric DNA, forming t-loops. We present evidence suggesting that t-loops can also form at interstitial telomeric sequences in a TRF2-dependent manner, forming an interstitial t-loop (ITL). We demonstrate that TRF2 association with interstitial telomeric sequences is stabilized by co-localization with A-type lamins (lamin A/C). We also find that lamin A/C interacts with TRF2 and that reduction in levels of lamin A/C or mutations in LMNA that cause an autosomal dominant premature ageing disorder--Hutchinson Gilford Progeria Syndrome (HGPS)-lead to reduced ITL formation and telomere loss. We propose that cellular and organismal ageing are intertwined through the effects of the interaction between TRF2 and lamin A/C on chromosome structure.