Intersection of languages in NP Can intersection of two languages in NP which are not NP complete be NP complete?

Can intersection of two languages in coNP which are not coNP complete be coNP complete?

Can intersection of two languages one in coNP but not complete and other in NP but not NP complete be NP complete or coNP complete?
Just an extended comment to better explain ARi's comment (I was writing it while I saw it).

It is sufficient to use a "large gap" approach similar to the one used in Lardner's theorem; for example:

$A_1 = \{ x \mid x \in SAT \land f(|x|) \text { is even}\} \cup \{x \mid f(|x|) \text{ is odd} \}$

$A_2 = \{ x \mid x \in SAT \land f(|x|) \text { is odd}\} \cup \{x \mid f(|x|) \text{ is even} \}$

Where $f$ is a slow enough increasing function computable in polynomial time. See for example its construction in Ladner's theorem proof in Appendix A.1 of Uniformly Hard Languages.

By construction $A_1, A_2$ are not NPC, but $A_1 \cap A_2 = SAT$