I am (re)reading Kunen's book on set theory, and I find myself curious to know what it means exactly for a structure to fail the comprehension schema.
In essence, it means every definable sub-collection of a set, is itself a set.
So if my understanding of the schema is correct, failing it would meant "there is a definable sub-collection of a set that is not a set", it sounds so oxymoron.
Also in that regard, are there any simple structures that demonstrate the failure of comprehension ?
If you change "is a set" into "is a set in our universe", maybe it becomes clearer. The meaning is the same, but it's a bit easier to interpret in exotic cases.
The class of (von Neumann) ordinals illustrate this nicely (they fulfill all the ZF axioms except comprehension, actually).
So assume our universe is the class of ordinals, and take any non-zero ordinal $\alpha$. Then $$ \{\beta\in \alpha\mid \beta\neq 0\} $$ is not in our universe.