In this paper we present an extensive analysis of the failure mechanisms of RGB (multi-chip) LEDs submitted to ESD testing: the tests have been carried out on several commercially available LEDs of three different suppliers. In order to better understand the failure mechanisms, we have submitted LEDs to ESD tests under reverse and forward bias condition separately, by means of a Transmission Line Pulser (TLP). The experimental results indicate that: (i) red LEDs (based on AllnGaP) have an higher ESD robustness with respect to green and blue samples (based on InGaN), both under reverse and under forward bias test; (ii) TLP negative pulses with a current smaller than the failure threshold can induce a decrease of the leakage current in GaN-based LEDs, due to a partial annihilation of defective paths responsible for reverse conduction: (iii) typical failure mechanism of devices is represented by a catastrophic event, with short-circuiting of the junction. Moreover, some of the analyzed red LEDs had shown "soft" failure, with gradual increase of the leakage current and corresponding decrease of the optical power, even without a catastrophic damage.