What is known regarding which hyperbolic groups are cubulated?


I take it the usual definition of cubulated is acting properly and cocompactly on a CAT(0)-cube complex.


My impression is that not all of them are, but I didn't manage to find references with a counterexample.


Are there known ways to create non-cubulated hyperbolic groups? 
Are there famous examples of non-cubulated groups?



Examples of non-cubulated hyperbolic groups
If a group $G$ satisfies Kazhdan's property (T), then any action of $G$ on a CAT(0) cube complex has a global fixed point. See Niblo and Roller's article *Groups acting on cubes and Kazhdan's Property (T)*. Examples of hyperbolic groups which satisfy this property include:


* Uniform lattices in quaternionic hyperbolic spaces.
* Random groups in Gromov's model for some density.


Also, by a theorem due to Gromov, any hyperbolic group admits a quotient which is hyperbolic and has property (T). 


