The X-files episode was from 2000. A 1979 short story by Gregory Benford entitled "Time Shards" used this very idea (the voice of a potter and another person conversing, being recorded in the clay pot), and the idea was discussed as early as 1969 in New Scientist.
It appears from several experiments by different people using typical line-incising techniques on clay pots, that sounds including human words could in fact be recovered:
"Acoustic Recordings from Antiquity", Proceedings of the IEEE 57 (8): 1465–1466, August 1969
Kleiner, Mendel; Åström, Paul (1993), "The Brittle Sound of Ceramics - Can Vases Speak?", Archeology and Natural Science 1: 66–72