Is it possible for batteries to give a false reading of being charged with a multimeter?
The batteries have increased in internal resistance due to aging. Since the multimeter does not draw significant current from the batteries, this internal resistance does not affect the reading significantly. The easiest way to validate this hypothesis this is to read the battery voltage while applying a load across the battery terminals, such as a resistor, so that a few tens of mA flow through the resistor. The multimeter reading should drop.

As batteries age, they tend to suffer a variety of deterioration modes, including pitting, oxide formation and scaling of the electrodes, as well as depletion of electrolyte. As the battery chemistry remains valid, the voltage across the cells will show more or less as expected. However, due to the deterioration mentioned, the ability to supply current reduces. This manifests as increased internal resistance, opposing the flow of current. 

In brief: Those batteries cannot fulfill the current demand any more, though the voltage under no-load condition remains as expected.