Are some power cables too old to be safe?
No, no 'Best By' date codes LOL.  However, your instincts are on track.  As the plastic jacket ages it becomes brittle and can crack allowing in moisture and presenting an obvious hazard.  However, if with inspection the jacket appears to be without defect; if it feels relatively flexible and supple then it may be ok. The cord ends need to be inspected as well.  Any signs of heat, burning or serious abuse chuck em. Thanks for being safety conscious.  P.