the CFL pumping lemma is not sufficient

It is tempting to treat the pumping lemma like a membership test: run it, and if the language passes, declare it context-free. That is WRONG, and the mistake matters. The lemma is a one-way street. It is a NECESSARY condition — every context-free language satisfies it — but NOT a SUFFICIENT one: some languages that are not context-free satisfy the pumping condition anyway. Passing the lemma proves nothing positive; only FAILING it proves a language is not context-free.

Logically, the lemma says 'context-free implies pumpable'. By contrapositive, 'not pumpable implies not context-free' — that is the valid direction you use in proofs. But the reverse, 'pumpable implies context-free', simply does not hold. So when a language pumps fine, you are stuck in limbo: it MIGHT be context-free or it might not be — the lemma has told you nothing either way. To settle such a case you need another tool: Ogden's lemma (which can fail where the plain lemma passes), intersection with a regular language to expose a non-context-free core, or a direct grammar-or-PDA construction to prove it IS context-free.

Concrete witnesses exist. There are languages designed so that every long string has an easy repeatable piece (so the plain pumping lemma is satisfied) and yet, on closer inspection with Ogden's lemma or a closure argument, they turn out not to be context-free. The lesson is the same as for regular languages: the pumping lemma is a non-membership detector, never a membership certificate. Treat 'it pumps' as 'inconclusive', and reserve 'it fails to pump' for your hard NO.