Why is my RGB not drawing more current?
It sounds like you have an inadequate PSU.


  "It's connected to a breadboard with a power supply that tests as giving 4.9 V and 2 A on a short circuit."


That's not how we check the output current. That tells you the current delivered when the output is 0 V. You want to know the maximum current delivered at 5 V. 

If you don't have the power supply specifications then load it gradually (with more and more load - lower resistance) until the voltage starts to drop. Then measure to current and you will have the maximum that the PSU can deliver. Note that this value may droop due to thermal limiting as the PSU heats up so it might be wise to assume 80% of that figure as the rated output.


  Looking at the specs for it (which I hadn't thought to do), it looks like it would have maximum output of 700 mA.


Most USB phone chargers will supply 2 A at 5 V. Get an old USB cable, cut the 'B' plug off and use the red (+) and black (-) wires.




  I've got 5 W resistors at 2 ohms for the green and blue leads (the datasheet says they have Vf = 3.6 V) and a 3.3 ohms for the red lead (Vf = 2.5 V).








simulate this circuit &ndash; Schematic created using CircuitLab

Figure 1. Schematics are so much better than words. This is what you've got. The voltmeters show the way to measure what's going on.

If your reading of the datasheets is correct then you should measure 2.5 V across R1 which would indicate that you have \$ \frac {V}{R} = \frac {2.5}{3.3} = 750 \ \text {mA} \$ through the red and \$ \frac {1.4}{2} = 700 \ \text {mA} \$ through the green and the blue. So my calculations agree with yours.

You may be losing some voltage elsewhere in your circuit due to high resistance wiring or a bad connection so check the voltage as shown with VM3 at the anode and at the bottom end of the resistor.