# Every cycle class a Chern class?

I am currently learning intersection theory of smooth algebraic varieties and I have the following question.

Let $$X$$ be a smooth projective variety and $$\mathcal{F}$$ a vector bundle on $$X$$. Then the $$i$$th Chern class of $$\mathcal{F}$$ is an element of the $$i$$th Chow group $$A^i(X)$$ of $$X$$. What about the converse? Can every class in $$A^i(X)$$ be realized as the $$i$$th Chern class if a vector bundle? Clearly, this is true for $$A^1(X)$$ and it is true for all $$i$$ if $$X$$ is the projective space. Is it true in general? If not, is it true for some nice varieties, for example for Grassmannians?

It is not true in general. The counterexample that I know is $$X$$ a general hyperplane section of $$LGr(3,6)$$ and the class of a line on $$X$$.