Vladimir Polkovnikov's "Gribok-teremok" is based on a story by cartoonist and children's author Vladimir Suteev. It's got a good moral lesson but a lot of the cartoon comes across as overly cute. The plot has a bunch of animals seeking shelter under a mushroom during a thunderstorm. It gets harder each time since there's less room, but the stakes get raised when a rabbit tries to escape a hungry fox.
I've liked most of the Soyuzmultfilm cartoons that I've seen (especially their adaptations of Winnie the Pooh), but this one was too fluffy. It was more like a 1930s Disney cartoon. If you authentically want to see it, it's available on Soyuzmultfilm's YouTube channel.