The Wiki entry also mentioned "two effects that lower the actual temperature needed", one being average kinetic energy and the other quantum tunneling "if [nuclei] have nearly enough energy." The term "nearly" isn't precisely defined though.
There isn’t a hard limit. But as you get farther away from the amount of energy that would be “enough” without tunneling, the probability of tunneling falls off exponentially.
Quantum tunneling effects are generally (maybe always?) exponentially unlikely, in the sense that the probability of tunneling will decay exponentially with the discrepancy between energy-available and energy-required.
