This is partly inspired by answers to the question: Question about Hodge number . Is there a family of compact complex manifolds, where the general fibres are Kähler, but for which $E_1$ degeneration of the Hodge to de Rham spectral sequence fails at the special fibre? Or, even better, such that the special fibre has nonclosed holomorphic forms?
I feel like I should know the answer, but somehow I don't. All the examples I know where the spectral sequence doesn't degenerate are nilmanifolds*, so they aren't even homotopic to Kähler manifolds by standard rational homotopy theoretic obstructions (e.g. they aren't formal). Also the famous Hironaka example [Ann. Math 1962] won't work either, because the special fibre is an algebraic variety, so the spectral sequence will degenerate (by an argument that can found in Deligne [Théorème de Lefschetz...]). Obviously, I haven't thought about this deeply enough, but perhaps someone else has**.
*I was bit sloppy yesterday, since the examples I have in mind include solvmanifolds. However, there are still topological obstructions to these being Kähler due to Nori and myself.
** From the answers, I gather that the work of Popovici suggests that there may be no counterexample.