Problem wiring a 3-way lamp switch
Yes, try another three-way lamp.

Three-way lamps have two filaments. One on for low, the other on for medium,and both on for high.

The screwshell is the neutral and the center button and outer ring are energized together for high. If you have a voltage tester, with the lamp switch on high,  you can carefully test from the center button to the outer shell. And then test from the ring to the outer shell. If you get 120 volts both times the switch is good. If not replace the switch.

If the switch is good it must be a bad lamp.

Good luck and stay safe!