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.
Expect to spend nearly 3 hours dismantling, testing, and running new wires in a chandelier. You'll need to know how to test for power and strip and connect wires. Before you begin, line up a helper to assist with removing the fixture and lay a drop cloth on a work surface to cushion the fixture as you work on it.