The most common problem is one or both conductors breaking internally at one of the ends due to continuous bending and stretching. Make sure the outlet is live - check with a lamp. Make sure any voltage selector switch is set to the correct position.
Move it back and forth a couple of times to make sure the contacts are clean. If the voltage readings check out for now, then wiggle the cord as above in any case to make sure the internal wiring is intact - it may be intermittent. Although it is possible for the adapter to fail in peculiar ways, a satisfactory voltage test should indicate that the adapter is functioning correctly. It's also possible that the power jack on the device itself is damaged from use or abuse.
If possible, confirm proper operation with a compatible adapter. With battery operated devices, there is usually a set of contacts that should close when the adapter is removed to connect the internal battery to the circuitry. If these don't operate properly, the device may not work off batteries (they may appear to not be charged), the AC adapter, or both.
Check the jack for obvious signs of damage (cracked, loose, etc.). A squirt of contact cleaner into the jack may clear up intermittent contact problems not due to actual damage.