The answer is yes, that cats do get stuck in trees and the reason they can't get themselves down is well established. More on that in a bit.
As to whether it happens often enough to warrent such a strong featuring in the public psyche is a subjective question and one which is difficult to answer. But the concrete "Does it happen at all" is much easier; a Freedom of Information Request was submitted to North Yorkshire Fire and Rescue Service enquiring How many cats that were stuck up trees, did you rescue in 2010 and the answer (as you'll see in the link above) was Seven.
Elsewhere, there has been a case of firemen having to fell a tree to rescue a stranded cat in Japan and a Tiger stuck in a tree at a zoo.
Most cats (Including most big cats, domestic cats) climb trees very easily due to the shape of their claws allowing them to scale upwards very easily. However to get down they usually jump from the lowest point they can reach, edge down backwards, or scamper down forwards.
Cats from the Genus Panthera and Leopardus are generally much better climbers and often live in trees, with some species (eg, the Margay) able to rotate their claws:
When descending a tree trunk, the margay can go down headfirst, but
does not have to scamper down quickly like the majority of cat species
must. The long claws, broad feet and nimble toes, plus the fact that a
margay has the ability to rotate its feet 180 degrees to an outward
position, make this possible. The margay is also agile enough to grab
a branch with its long claws to prevent a serious fall when it does
make a mistake while in the tops of trees