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Beam Me Up, Scotty!: Contrary to popular criticisms, a few Barney fans debunked the rumor that Barney described strangers as being "friends you never met."note This rumor started from a post to a Usenet forum called Alt.Barney.Dinosaur.Die.Die.Die Instead, the very opposite was stated by Derek in the episode "Playing It Safe" (he said that they're not only people you don't know, but that "they could be bad people."), which led to a short Too Smart for Strangers moral using a play of Little Red Riding Hood and a song (in traditional Barney fashion) to reinforce the moral of not talking to them.
Dawson Casting: Some of the kids are well into their preteen years by the time they leave the show. Chip was a notable case. Luci looked a lot older than the other kids.
Of course, Luci was Tina's older sister, so it was semi-justified.
In Brazilian Portuguese dub, Beakman is the voice of Barney.
It Will Never Catch On: Prior to the TV series and videos becoming hits, Barney's producers were faced with this exact trope on a number of occasions.
Connecticut Public Television executives felt this way when they couldn't see the series' appeal after watching a Barney video. After they showed the video to kids and saw their reactions, however, the tides turned and CPTV immediately got a TV contract for the Barney series.
Song writers Lory Lazarus and Stephen Baltes, who wrote songs for the initial three Barney & The Backyard Gang videos, were told by friends to leave the company for this reason. They came back to write for Barney & Friends' third season, once the character's popularity took off.
The Barney suit was originally going to be somewhat more "realistic" looking, with scales, claws, pointed teeth, and even lacrimal ridges. Through the various drafts, the creative team told costume designer Irene Corey that the costume would be too frightening for kids, until it came to what would become Barney's eventual design. Proof: 
A young Kyla Pratt plays Marcella, one of the children.
One of the dancing waiters in the restaurant scene is none other than Josh "Ted Mosby" Radnor. Also doubles as Hilarious in Hindsight because it would not be the last time we'd see Josh Radnor and the name "Barney" together.