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I'm afraid not. :( That's a good suggestion, but from what I remember, this wasn't really an educational short or feature film, but basically a film showing several examples of cute animals being, well, cute. Okay, well, I can't really add to descriptions about the movie itself, but if it helps, I can add a bit about the site that reviewed it: it was a site where a good deal of the reviews (all of them text reviews) were on horror movies, like a multi-part abridged history of the Nightmare on Elm Street movies, an article talking about some of the various Phantom of the Opera film adaptations (with one bit of that pointing out Lon Chaney's rather hilarious Oh Crap face as the Phantom when it seems like Christine is going to choose the locust, blowing all of them up), and a review of Teeth, with one of the screencaps placed among the text being of a crab carrying off the severed penis of one of the guys who tried to rape the protagonist (with said penis being censored).
Okay, I found it! The name of the film is "Bill and Coo", and it's apparently a movie with an actual narrative, albeit a very threadbare one that's just an excuse to show off cute birds. Plus, it's apparently from the late 40's, not the 50's or 60's. http://www.imdb.com/title/tt0039188/
Thanks again to everyone for their suggestions and help!

Okay, I remember seeing a rambling, angry, baffled text review of this film from either the 50's or 60's that was basically a huge compilation of pet birds (like parakeets, lovebirds, parrots, and so on) doing cute tricks and stuff. Like, it doesn't really have a plot, it's just a compilation of pet birds and I think a few other animals either doing tricks and/or being dressed up in little costumes. Does anyone know what that film is?