With the Light Step perk, you'll never set off an enemy's mines or floor-based traps.
Traps can still be set off by companions (including "floating" companions such as Sergeant RL-3 or ED-E), possibly resulting in the player character taking damage as well. Only the companions will take damage from rigged shotguns (unless one is in between) and bear traps. Other traps, such as explosive baby carriages, still go off if approached, and grenade bouquets triggered by a floor trap will still go off by making contact with the grenades, or by shooting them.
Maybe it was the landmine that almost blew your leg off, or maybe it was the sight of your buddy in pieces next to you. Whatever the cause, when you meet the triggering criteria for a landmine, door bomb, or steel trap, there is a 4 in 20 chance per rank that it fails to go off anyway.