Often what happens is that the common wire (the 'Earth ' or negative wire) which is common to many items in a car and usually connected to the car's chassis, does not make proper contact because of dirt, loose connection or rust. And even though you can obtain a voltage reading, under load and drawing current, the 'rusty' connection may present such a large resistance that there is just not enough current to drive the item under question. Note that a 50W car lamp can easily draw 4A plus and even though LEDs draw small currents, they are used in large numbers and are of the hi- bri type needing 20-50/70mA each.
Thus, try cleaning/removing rust, dirt & grease from connections associated with the item, then solidly tighten the connection(s).