I have a '95 Jeep Cherokee Sport. About a month ago, both high beams went out at once. The dashboard indicator displays the high beam on light, but no lights. After checking with a multimeter, I discovered the lamps are fine, but power is not being supplied to the lamps. All fuses and relays are in good shape. Is it possible the connection is bad, could it be the wire, or something else?

 

Check the multi-function switch and consider installing a harness.

 

The high beam indicator light turns on in the dash board. According to the wire diagram this means that the issue is after the switch, and before the headlamps themselves.

 

There are no relays involved. Power is routed directly from the headlight switch to the high/low (multifunction) switch, then to the bulbs. Either you have a broken wire or a failed multifunction switch.
It's possible a failing switch could pass enough current to light the dash indicator but not the high beams.

 

Bingo.


Also, corrosion or other poor connection at any point may inhibit the flow of current. This may pass voltage so that a meter shows the correct reading, but when a load is applied (like your headlights), the voltage drops, sometimes to effectively zero.

Try your tests again with a test light instead of a meter.