A concept of dynamical coherence I'm trying to make an overview of the study of partial hyperbolicity and there is an interesting concept of dynamical coherence which appears there. Some call it mild (see the Thesis of Pablo Carrasco, Compact Dynamical Foliations 2010), some call it strong and unnatural (see the work of Amy Wilkinson and Keith Burns Dynamical coherence, accessibility and center bunching). The definition which is the most common is that local cental-unstable $E^{cu}$ and center-stable $E^{cs}$ bundles integrate to foliations $W^{cu}$ and $W^{cs}$. 
Let us suppose, that in a normally hyperbolic case, i.e. when we already have the $E^c$ that integrates to a foliation F, at which some diffeomorphism is hyperbolic.
My question is how the normally hyperbolic (i.e. partially hyperbolic on foliation) system could be dynamically incoherent and is this concept somewhat related to the concept of local product structure? 
My question is, what is a simplest example of a normally hyperbolic foliation when $E^cu$ and $E^cs$ do not integrate to foliations? And how "often" does it happen in the world of normally hyperbolic foliations? 
PS. Updated after a useful remark of Rafael Potrie, the definition of a dynamical coherence is now more precise.
 A: In general, it is not known if a partially hyperbolic diffeomorphism should be dynamically coherent. 
There are two obstructions for integrability of the center bundle. One is that the distributions are not integrable (Frobenius conditions fails) and the other one is that the distributions may lack of diferentiability (and so uniqueness of integrability may fail). 
For the first obstruction, Wilkinson noted that in diffeomorphisms with high dimensional center (i.e. Anosov automorphisms on nilmanifolds) the bracket condition fails. 
In the absolute partially hyperbolic setting, for diffeomorphisms of $T^3$ dynamical coherence has been obtained by Brin-Burago and Ivanov (http://www.pdmi.ras.ru/~svivanov/papers/coherence.pdf). This has been extended to nilmanifolds by Hammerlindl and Parwani (http://arxiv.org/abs/1103.3724, http://arxiv.org/abs/1001.1029). 
For pointwise partially hyperbolic systems (a weaker condition), recent examples have been constructed by Rodriguez-Hertz, Rodriguez-Hertz and Ures showing that dynamical coherence may fail even in $T^3$. On the other hand, I have recently proved that if the partially hyperbolic diffeomorphism of $T^3$ is transitive (or volume preserving) then it must be dynamically coherent (see http://www.mat.puc-rio.br/edai/textos/potrie.pdf).
Local product structure I believe has something to do with plaque-expansiveness, which allows one to show robustness of dynamical coherence thanks to the work of Hirsch-Pugh and Shub, but in general it is not enough to show the existence of an invariant foliation tangent to the center.  
