XIST RNA established the precedent for a noncoding RNA that stably associates with and regulates chromatin, however it remains poorly understood how such RNAs structurally associate with the interphase chromosome territory. I demonstrate that transgenic XIST RNA localizes in cis to an autosome as it does to the inactive X chromosome, hence the RNA recognizes a structure common to all chromosomes. I reassess the prevalent thinking in the field that a single protein, Scaffold Attachment Factor-A (SAF-A/hnRNP U), provides a single molecule bridge required to directly tether the RNA to DNA. In an extensive series of experiments in multiple cell types, I examine the effects of SAF-A depletion or different SAF-A mutations on XIST RNA localization, and I force XIST RNA retention at mitosis to examine the effect on SAF-A. I find that SAF-A is not required to localize XIST RNA but is one of multiple proteins involved, some of which frequently become lost or compromised in cancer. I additionally examine SAF-A’s potential role localizing repeat-rich CoT-1 RNA, a class of abundant RNAs that we show tightly and stably localize to euchromatic interphase chromosome territories, but release upon disruption of the nuclear scaffold. Overall, findings suggest that instead of “tethering” chromosomal RNAs to the scaffold, SAF-A is one component of a multi-component matrix/scaffold supporting interphase nuclear architecture. Results indicate that Cot-1 and XIST RNAs form integral components of this scaffold and are required to maintain the chromosomal association of SAF-A, substantially advancing understanding of how chromatin-associated RNAs contribute to nuclear structure.