I have 4 recessed lights on a single switch/circuit. One of them stopped working. I assumed that the bulb went bad, but have replaced it with multiple known-working bulbs. Also confirmed that the center contact makes good contact with the bulb. The socket is connected to the box with armored cable. Could something in the box itself have gone bad? How would I trouble-shoot this?
Recessed can lights have several issues you have already checked the center contact, now there are 2 other things to check. First I would look for a thermal overload, these are a small sensor that is inline with the hot conductor to the center tab on the light socket, if a larger lamp was used than the fixture was rated for this is usually the cause, the overload is temp sensitive and opens when hot, these sensors don’t last long if a over sized lamp was used in the fixture they are usually small some about the size of a pea, many about an inch long and smaller than a pencil in diameter, to check this the circuit needs to be turned off and an ohm meter from center contact to the black wire should read close to zero if open the overload has failed and needs to be replaced (sometimes cheaper to replace the fixture. The other thing I find on older fixtures is the heat for all the years has baked the wiring and with some vibration one of the conductors opened at the fixture or the splice to the power that should be in a box ether on the fixture or connected close by.