Why can't I use Ohm's Law on my circuit?
Ohm's law applies to the resistor. The voltage across the resistor will be about 3V. The remaining 2V or so appears across the LED. 

To a rough approximation I = Vr/R ~= (5V-Vf)/200. Vf, the voltage across the LED, stays more or less the same at different currents within the normal operating range. 

So if you increase the voltage to 6V you would expect the current to increase to about 20mA (actually a bit less because the voltage across the LED will increase a bit). 

From this answer, there's a graph that is similar for most red LEDs. 

As you can see, at about 15mA Vf is a bit under 2V. At 20mA, Vf is just about 2V exactly. At 30mA it's more like 2.05V.