I don't see how. Curl and divergence are essentially "opposites" - essentially two "orthogonal" concepts. The entire field should be able to be broken into a curl component and a divergence component and if both are zero, the field must be zero.

I'm visualizing it like a vector in $\mathbb{R}^2$. A vector cannot have a zero $x$ component and a zero $y$ component and still be non-zero.

EDIT: Here's a slightly more formal formulation of my thoughts: The way I see it, the curl and divergence form a "basis" - they are essentially orthogonal vectors. So how can a non-zero vector not be in their span?

**Please don't just give me a counterexample. Please explain why my logic is incorrect**.