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.