Often with light fixtures, each light has a pair of wires that come up and out of the fixture. You wire all of the hots together with the line hot, and all of neutrals together with the line neutral. If the wire nuts that connect all of the wires were not installed properly, it is possible that one of the wires is no longer making a solid electrical connection. It is also possible that one of the wires is damaged.
I would suggest that you remove the light fixture and examine all of the connections. Look for loose wires (you should not be able to easily pull them out of the wirenut), nicks, burns, etc.
If everything is visually OK you should test each wire to each socket with a multimeter. Test for continuity by ensuring resistance goes close to 0 when you make a connection. Repeat this until you find the problem. Make sure the power is off when you do this.
Your Plan
Your analysis about two wires makes an assumption that is forbidden in residential electrical work. One of the wires in a standard fixture is the neutral whose state cannot change. It must always be connected to the other neutrals in the system, connected to the main electrical panel and ultimately to ground. If 1 reresents connected, the white wire is always in the 1 state.
The black, or hot wire can be connected or not. Similarly, in a three wire system, the red wire, as well as the balck, can also be used as a hot wire and can be connected or not, depending on the state of switches. In a three wire system, the white still stays connected all the time.
How to do it
If you want separate controls, you need a separate wire to power (carry the hot current) to each bulb. A single neutral (white) can be used.
Hot wire originating at switch
If the power is presently comming into the switch box from the electrical panel, and then being routed to the fixture, the standard way to wire this would be to run a three wire cable (white/black/red) from the switch to the fixture and to use a double switch, such as this type.
The white wire coming from the panel is connected to the white wire going to the fixture. The black wire coming from the panel is connected to the incomming sides of both sections of the switch. The black wire to the fixture is connected to one of the outgoing terminals and the red wire from the fixture is connected to the other outgoing terminal (outgoing because the power is coming through the switch and going out to the fixture).
Hot wire originating at fixture
If the power comes directly to the fixture, and there is just a two wire cable going to the switch, you need to run a three wire cable to the switch. The white wire will actually be used as a black/hot wire to carry power to the switch. It needs to be marked with black tape a both ends to show this. At the fixture this pseudo black wire is connected to the black hot from the main panel. The real black is connected to one bulb and the red to another bulb.
At the switch, the white (now marked black) is the incoming line to both switches and the red and black are attached separately to the outgoing terminal on each switch.
In either setup, the switches are independent. You can turn on either, both or neither.
There are double throw switches that would allow you to have one or the other or neither, but not both. You still need the three wire cable at the switch site.
Best Answer
Since you have power you must not have a return path or neutral that is making contact. In these cases a real volt meter is helpful, but checking the neutral at the panel and any connection points you may visually spot the damage, as I did today on a circuit that was not working. I pulled the device out and the neutral wire had burned off. If you can trace the cable from the panel, make sure if it goes through any outlets you check them; also, it could be any place on the run from the panel to the light that is causing the problem.