Suppose $X$ is a complex manifold.

If $X$ is Kahler, the cohomology groups decompose into subgroups represented by $(p,q)$ forms.

If $X$ is not Kahler, I think the decomposition may not hold?

Is there an example where we have a nonzero class be represented by both a $(p,q)$ form and a $(p',q')$ ($(p,q)\neq (p',q')$)form?