The majority of relays installed on aircraft will have a diode installed in parallel with the relay coil. This diode is often referred to as a "Suppression Diode". The diode can be installed internal as part of the relay assembly or external as a standalone component. The purpose of the diode is for circuit protection.
All inductors (relay coils included) will build up an electrical field around themselves when voltage is present. If the voltage is removed the field will collapse. As the field collapses a current is induced in the wiring connected to the coil. The suppression diode gives a path for this current to dissipate.
The diode is installed in the "Reverse Bias" position to the control voltage. Current cannot flow through it. All the voltage will be present for the relay coil to energize. The circuit will remain in this condition until the control voltage is removed. With the voltage removed, the coil field will collapse. The induced current is in the "Forward Bias" relationship to the diode, so it basically reduces the potential back across the coil. No current will be felt in the control voltage input wire, reducing the possibility of computer or component damage.
The suppression diode must be accounted for when continuity checking a circuit that has a relay coil in it. It is very possible to read through just the diode when the relay coil is open. Isolation of the diode might be needed for verification of a good circuit.
If it's internal, the only way you would know about the diode was if the relay coil was open. A complete relay failure would certainly be noted in whatever aircraft system it was a part of.
There's really no way to verify if a internal diode is bad (in or out of the circuit), just chunk it and install a new one.