Originally, the character of Snuffleupagus only ever interacted with Big Bird. He'd always come and go when no one else was around, and consequently no one ever believed Big Bird when he told them of his existence. The producers decided to reveal him to the other characters partially because they felt it was sending a bad message to children that adults will not believe them if they have something important to tell them.
Strangely, despite appearing in Big Bird's flashback, Olivia was the only recurring adult character at the time who did not appear in the episode, even during Snuffy's big reveal.