The feast of Epiphany (originally observed on January 6 in the Roman Rite), is one of the oldest liturgical celebrations, even before Christmas was established on December 25.
The name itself refers to the "manifestation" of Jesus during 3 particular biblical events.
Pope Benedict XVI refers to this tradition in a homily on Epiphany in 2009.
Dom Prosper Gueranger in his Liturgical Year states that it was the Church's tradition for many centuries that these three events occurred on the same day, January 6.
Gueranger explains the origins of this tradition and concludes with, "it is impossible to prove that the sixth of January was not the day [for any of these events]. For us the children of the Church it is sufficient that our Holy Mother has assigned the commemoration of these three manifestations for this Feast, we need nothing more to make us rejoice in the triple triumph of the Son of Mary."
Basically, we can't know for certain whether or not these events happened on the exact historical day of January 6. Yet, we can still recollect them on the feast of the Epiphany and rejoice in the manifestation of the Lord to the world.
