For many years, Big Bird was the only character on the show who saw him (he only came along when Big Bird was alone). The other characters teased Big Bird when he said he had seen the Snuffleupagus, because they didn't believe there was such an animal.
This was modeled in part on the imaginary friends some young children have. On November 18, 1985 the show was changed so other characters saw Snuffleupagus, to avoid having adults tell a sympathetic character that he was making things up. The creators were also worried about kids thinking that others might not believe them.
In Elmo Saves Christmas, it is revealed that Snuffy's grandmother lives in Cincinnati, Ohio.
The Goodbye episodes (1980s)
The adult characters became sick and tired of Big Bird always using his "imaginary friend" Snuffleupagus as a scapegoat for anything that goes wrong when they're away. One episode has them urging Big Bird to say his good-bye to Snuffy once and for all; they are driven to the brink with all the nonsense.
Even the newspapers on Sesame Street read with front page headlines saying that Snuffy's got to go. So, Big Bird and Snuffy sadly say their good-byes in Big Bird's nest area. They cry in remorse of their forced separation. Snuffy's tear falls onto Big Bird while they hug, leading Big Bird to the conclusion if he can even feel the water from the tears, Snuffy must be real.