The concrete example usually used is the fence/fenceposts.

Four fenceposts (including the beginning and the end) are needed to divide a fence into three sections. Unless, as pointed out, it's circular fence, in which case the beginning and the end are identical.

I've seen tertiles, quartiles, quintiles, noniles, and deciles used in medical or psychology papers of varying degrees of scholarship (and/or pretentiousness). Often the choice seems to be driven by what measurement will render the conclusion "significant." Caveat lector. [Did I get that right?]