This paper demonstrates that when InGaN LEDs are submitted to a constant reverse bias, they can show a time-dependent breakdown, that leads to the catastrophic failure of the devices. By submitting green and blue LEDs to constant voltage stress in the range between -40 V and -60 V we demonstrate that: (i) under reverse bias conditions, current is focused on localized paths, whose positions can be identified by electroluminescence measurements, and that originate from the presence of extended defects; (ii) during a constant voltage stress, the reverse current of the LEDs gradually increases; (iii) for longer stress times, all devices show a time-dependent breakdown; (iv) time-to-failure has an exponential dependence on stress voltage, and is Weibull-distributed.