A curable composition contains (A) a polyorganosiloxane base polymer having an average per molecule of at least two aliphatically unsaturated organic groups, optionally (B) a crosslinker having an average per molecule of at least two silicon bonded hydrogen atoms, (C) a catalyst, (D) a thermally conductive filler, and (E) an organic plasticizer. The composition can cure to form a thermally conductive silicone gel or rubber. The thermally conductive silicone rubber is useful as a thermal interface material, in both TIM1 and TIM2 applications. The curable composition may be wet dispensed and then cured in situ in an (opto)electronic device, or the curable composition may be cured to form a pad with or without a support before installation in an (opto)electronic device.