Suppose $X$ is a complex manifold.

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

If $X$ is not Kähler, 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 $(p',q')$-form with $(p, q) \neq (p',q')$?