Deformations of Kähler manifolds where Hodge decomposition fails?  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**.
Footnotes 
*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.
 A: If any example exists, then the general fibre of the family cannot be projective. 
In fact, Dan Popovici ["Limits of projective manifolds under holomorphic deformations", arXiv.09102032] recently proved the following 
Theorem. Let $\pi \colon \mathcal{X} \to \Delta$ be a complex analytic family of compact complex manifolds such that the fibre $X_t:=\pi^{-1}(t)$ is projective for all $t \neq 0$. Then $X_0:=\pi^{-1}(0)$ is Moishezon.
Since Moishezon manifolds admit a projective algebraic modification, it follows that their Hodge-Frolicher spectral sequence degenerates at $E_1$. In particular, Hodge decomposition holds for $X_0$. Notice that in this case $X_0$ is Kähler if and only if it is projective. 
A: Just as a comment.
In general, given a family of sG (i.e., strongly-Gauduchon) manifolds, the limit need not to be still sG: an example was provided by Ceballos, Otal, Ugarte, and Villacampa, http://arxiv.org/abs/1111.5873 (see also http://arxiv.org/abs/1210.0406).
It may be possible that the sG property is preserved to the limit assuming stronger conditions on the fibres: however, the published version of the paper by Popovici does not prove this fact, as noticed also by YangMills.
On the other hand, an example of a family of (non-Kähler) manifolds satisfying the Hodge decomposition (namely, the $\partial\overline{\partial}$-Lemma) and whose limit does not, is studied in http://arxiv.org/abs/1305.6709.
Note that there are several examples of sG manifolds non-satisfying the Hodge decomposition: for example, every nilmanifold admitting a balanced or sG metric.
A: This is known, for projective (even Moishezon) 
manifolds as shown by Dan Popovici in his 
paper http://arxiv.org/abs/1003.3605
For general Kaehler manifold, this is conjectured.
Popovici has proved that a property of "strong Gauduchon"
is preserved in limits  http://arxiv.org/abs/1009.5408
and (I think) there are no example of strong Gauduchon
manifold without Hodge decomposition.
