Is the Conjunction of a necessary condition with a sufficient condition, a necessary and sufficient condition?

I came across a question where i had to find the necessary and sufficient condition for a property $$p$$. So i thought of finding a necessary condition $$n$$ and a sufficient condition $$s$$ separately and then putting a conjunction between them. But I'm not sure whether that would give me the necessary and sufficient condition. I'm not able to think of a counter example to this at the moment. Formally my query is,

Are the sequents $$(p\Rightarrow n) \land (s\Rightarrow p) \vdash (n\land s) \Leftrightarrow p$$ and $$(n\land s) \Leftrightarrow p\vdash (p\Rightarrow n) \land (s\Rightarrow p)$$ valid?

No, because the sufficient property could be false: if $$n=p=\top,s=\bot$$, then $$p⇒ n$$ and $$s⇒p$$ are both true, but $$n\wedge s=\bot$$ while $$p$$ remains true, so the equivalence certainly cannot hold.
One could see that on an intuitive level by noting that since $$s\implies p\implies n$$, the statement $$s\wedge n$$ only depends on the statement of the stronger of both, namely the sufficient condition $$s$$. But that would reduce the right side to $$s\Leftrightarrow p$$, which is absurd in general.