Portables: L-shaped MagSafe power adapter does not charge computer and LED indicator light does not illuminate

In certain situations when you connect a new 60W or 85W MagSafe power adapter to the MagSafe port of an older MacBook or MacBook Pro computer, the adapter may not charge the computer and the LED indicator light on the MagSafe connector of the adapter may not illuminate. An example of an L-shaped MagSafe power connector and adapter: