Older chandeliers often need repair. Many were manufactured with little regard to the heat the bulbs produce, and larger bulbs than recommended have often been installed. Overheating makes the wire insulation brittle. A typical fixture has cord running through tubes to the bulb sockets, allowing several opportunities for malfunctions.
Because sockets and the wires attached to them are near a hot bulb and often enclosed, they deteriorate. If all the lights do not work, the stem wire probably needs to be replaced. If some wiring needs to be replaced, consider rewiring the entire fixture -- it won't take much longer.
If only one light fails to come on, try pulling up the contact tab inside the socket. Vacuum dust from the socket. If the bulb still does not light, remove the socket and test it.
A chandelier is often suspended by a chain, which must be securely anchored to the box hardware. The main wires run from the box down through the chain to a junction box. There they connect to wires that lead to individual light sockets.
About 3 hours to dismantle, test, and run new wires in a chandelier
Phillips screwdriver, voltage tester, continuity tester, wire strippers, long-nose pliers
Testing for power and continuity, stripping and connecting wires
Line up a helper to assist with removing the fixture. Lay a drop cloth on a work surface to cushion the fixture as you work on it.
Cord wire, electrician's tape
Shut off power to the circuit (in addition to flipping off the light switch). Support the chandelier. Loosen the screws or screw collar holding the canopy in place and slide it down. Pull the wires apart, restore the power temporarily, and test for power. Shut off the power again.
Attach a continuity tester clip to the metal threads inside the socket and touch the neutral (silver) terminal with the probe. Then clip onto the brass screw and touch the probe to the contact tab inside the socket. If the tester light does not come on for both tests, replace the socket.