Jarris,
They can light up independently if the leads are shorted to the heat sink, but when you power them up, the power could be going from the positive pad of the first Cree and going straight through the heat sink to the negative pad of the last. This can allow one to not light up. Do you have a voltage meter that can test continuity? If so, try seeing if any of the positive or negative pads shorted tout to the heat sink. That can happen easily, even when using AA thermal epoxy. If you do not have a multimeter with that function, get 1 or 2 AA batteries and hook it up to a spare low voltage light bulb (maybe from the spare from a M*g 2D or 2C). Then touch the remaining two open wires where you need to test continuity (one on the lead pad of the Cree and the other on the heat sink). If it lights, then you have a short. This should be a low enough voltage set up where you cannot do any damage to the LEDs.
I would try this before you start applying that much power. If the LEDs are shorted in a way where two are attached in parallel, then you can do damage since two in parallel will run at a max of only 7.8V. If you are only attaching only 3.7V to them all and only two are lighting up, then it is NOT because of not enough voltage! 3.7V can barely light up two white LEDs in series, let alone three (if there is not a short). Even though they do light up at low voltages, this 3.7V should not be enough to light up three, even dimly. If totally unsure and you just want to hook up more voltage, go with something like a battery pack that is around 7 volts, or maybe even use a little 9V battery. If two light up brightly, and not the middle one at all, then you need to check for a short for sure. I would hate to see you damage your $25 emitters, so I would slowly work your way up with testing (continuity, ~7 to 8V, then ~12V when confident).
Well, do not let this get you down. This stuff happens to all of us. Good luck with the testing. Please keep us updated. We are here to help you...
-Tony