The Axiom of Choice (AC) implies that every set can be well-ordered, so every infinitecardinality is an aleph; but in the absence of AC there may be sets that can't be well-ordered (don't posses a bijection with any ordinal) and therefore have cardinality which is not an aleph.

These sets don't in some way sit between two alephs; they just float around in an annoying way, and can't be compared to the alephs at all.

No ordinal possesses a surjection onto such a set, but it doesn't surject onto any sufficiently large ordinal either.