The JTE is usually stronger where the electron density associated with the degenerate orbitals is more concentrated. This effect therefore plays a large role in determining  the structure of transition metal complexes with active internal 3d orbitals. 
The most iconic and prominent of the JT systems in coordination chemistry is probably the case of Cu(II) octahedral complexes. While in perfectly equivalent coordination, like a CuF complex associated to a Cu(II) impurity in a cubic crystal like KMgF, perfect octahedral (O) symmetry is expected.  In fact a lower tetragonal symmetry is usually found experimentally. The origin of this JTE distortion it revealed by examining the electronic configuration of the undistorted complex. For an octahedral geometry, the five 3d orbitals partition into t and e orbitals (see diagram).  These orbitals are occupied by nine electrons corresponding to the  electronic configuration of Cu(II). Thus, the t shell is filled, and the e shell contains 3 electrons. Overall the unpaired electron produces a E state, which is Jahn–Teller active.  The third electron can occupy either of the orbitals comprising the e shell: the mainly  orbital or the mainly  orbital.   If the electron occupies the mainly  level, which antibonding orbital the final geometry of the complex would be elongated as the axial ligands will be pushed away to reduce the global energy of the system. On the other hand, if the electron went into the mainly  antibonding orbital the complex would distort into a compressed geometry. Experimentally elongated geometries are overwhelmingly observed and this fact has been attributed both to metal-ligand anharmonic interactions  and 3d-4s hybridisations. Given that all the directions containing a fourfold axis are equivalent the distortion is equally likely to happen in any of these orientations. From the electronic point of view this means that the  and  orbitals, that are degenerate and free to hybridise in the octahedral geometry, will mix to produce appropriate equivalent orbitals in each direction like  or .
The JTE is not just restricted to Cu(II) octahedral complexes. There are many other configurations, involving changes both in the initial structure and electronic configuration of the metal that yield degenerate states and, thus, JTE. However, the amount of distortion and stabilisation energy of the effect is strongly dependent on the particular case. In octahedral Cu(II), the JTE is particularly strong because
# the degenerate orbitals display a strongly antibonding σ character 
# Cu is a transition metal with a relatively strong electronegativity yielding more covalent bonds than other metals which allows to increase the JT linear coupling constant.
In other configurations involving π or δ bonding, like for example when the degenerate state is associated to the t orbitals of an octahedral configuration, the distortion and stabilisation energies are usually much smaller and the possibility of not observing the distortion due to dynamic JT effects is much higher. Similarly for rare-earth ions where covalency is very small, the distortions associated to the JTE are usually very weak.
Importantly, the JTE is associated with strict degeneracy in the electronic subsystem and so it cannot appear in systems without this property. For example, the JTE is often associated to cases like quasi-octahedral CuXY complexes where the distances to X and Y ligands are clearly different. However, the intrinsic symmetry of these complexes is already tetragonal and no degenerate e orbital exists, having split into a (mainly ) and b (mainly ) orbitals due to the different electronic interactions with axial X ligands and equatorial Y ligands. In this and other similar cases some remaining vibronic effects related to the JTE are still present but are quenched with respect to the case with degeneracy due to the splitting of the orbitals.