Yes - the faithful may confidently believe in the tradition of the Brown Scapular Apparition.  Although we do not have ample documentation in the wake of the 1251 A.D. apparition, things changed following it.  The Carmelite Order received full recognition and stability from the Church, became unified, and vocations increased.  Even the single-piece white mantle was cut down the middle to make the Scapular visible.  

A striking confirmation of this tradition happened during the last apparition of Our Lady of Fatima on October 13th, 1917, when Mary appeared dressed in the Carmelite Habit as Our Lady of Mt. Carmel, holding out the Brown Scapular.  The seer, Venerable Sister Lucia, would later say that Our Lady was asking all to wear her Brown Scapular, and that the rosary and the scapular go hand in hand.  Lastly, and most importantly, the authority of the Church has blessed this tradition through the centuries.