A 'collection' of elements which does not form a set

Yes, this is possible. One of the ways to see this is the idea of forcing as a tool for adding new sets to the universe.

The first example is adding a new set of integers. As the new set is made only of integers, it is "a collection of integers" but it is not a set in the model. It is not a class either, though. It just exists in a larger universe.

The problem here is that collections might hold information that contradicts with the rest of the axioms of ZF(C). For example, if $M$ is a countable model of ZFC, then there is a bijection between $M$ and the natural numbers (of $M$), which means that there is a collection of integers which codes the whole model.

If we add that collection as a set, then $M$ would be able to recognise itself as a set. Or at least it should. But it really can't. So you will have to violate the axioms of ZFC if you add that as a set.

Finally, it is important to point out that when we have this sort of situation where there is a collection which is a sub-collection of a set, but not a set, then this collection is not definable, and there is no way for the model you're working in to recognise it. In particular, it is not a class either. We are only able to talk about these collections from outside the model.

You can define, for example, a "class", which is formally nothing more than a predicate, but which we think of "the things which satisfy that predicate". The axiom schema of comprehension tells us that certain classes are in fact sets (namely, if the class is defined as a subclass of a set).

There is a class of all sets: one appropriate predicate is simply "true". The empty set is a class: for example, a predicate would be "false". There is a class of all finite sets: such a predicate would be "there is a bijection from $X$ to some finite ordinal".