This is partly inspired by answers to the question: .
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.