The Jahn-Teller effect is one of the most fundamental phenomena important not only for physics but also for chemistry and material science. Solving the Jahn-Teller problem and taking into account strong electron correlations we show that quantum entanglement of the spin and orbital degrees of freedom via spin-orbit coupling strongly affects this effect. Depending on the number of d electrons, it may quench (electronic configurations t 2 2g , t 4 2g , and t 5 2g), partially suppress (t 1 2g), or, in contrast, induce (t 3 2g) Jahn-Teller distortions. Moreover, in certain situations, interplay between the Jahn-Teller effect and spin-orbit coupling promotes formation of the "Mexican hat" energy surface facilitating various quantum phenomena.