Can someone untreated for HIV start to show symptoms after fifteen years.?

Yes. The period of latency in which an infected person doesn't have symptoms is unique to each person. It is possible that someone could be asymptomatic for many years prior to becoming ill with an infection or other aids defining diagnosis.
YES. The virus can slowly destroy the cd4 lymphocyte cell mediated immunity so that many years after HIV infection opportunistic infections such as pneumocystis pneumonia develop.