One possibility is that there's a high-resistance point in the circuit. Your multimeter places almost no load on the circuit, so the voltage drop through the high resistance is negligible. With the bulb in the socket, however, the voltage drop will be much higher, and little current will flow through the bulb. If you can measure the voltage across the bulb with the bulb in the socket, or otherwise in the circuit when you measure the voltage, you can see this effect.
Cause of the high-resistance point could be corrosion in the wire or at a connection somewhere in the circuit.