Don't listen to Harvey, wire can handle less power when running 12 volts than it can when running 120. For example, 14 gage romex is specified to handle 15 amps; at 120v that would be 1800 watts of power. This is a large but acceptible draw for a 15 amp 120v circuit. Now try pushing that same power on a 12 volt circuit. 1800 watts / 12 volts = 150 amps! Whoops, you've just fried your cable.
A good rule of thumb is that a 12v circuit can only handle 1/10 the power of a 120v circuit. So your load should not be more than 180 watts for 14 ga romex.
I don't know your qualifications to be spouting like that, but you're a bit off the point. His question was whether the Romex he has in place will meet his requirements. An LED array can require as little as 10 -12% of the power required to produce the same luminoscity as a typical incandescent bulb and about half of the power to obtain the same luminoscity as a CFL.
The actual current through the Romex to the LED array will be determined as much by the design of the system, including the power supply (output voltage, and whether it's switching, linear), any dimming system and whether the LED's are connected in series, parallel or a series/parallel configuration.
You're right that the issue is current draw through a given length of wire because even 6 nines pure silver has some finite resistance.
Using plumbing as a metaphor, voltage is the equivalent of water pressure, current is the equivalent of the rate at which water flows through a pipe, and resistance is the equivalent of the diameter of the pipe. When you try to shove too much water through a pipe of a given diameter, you get back pressure. When you try to shove too much current through the resistance of a wire, you get heat. The only difference between a wire and a fuse is that the fuse is designed to melt and open when too much current passes through it.
He has professional electricians helping and advising him. If they're licensed and bonded, I would think they would be aware of the safe range of operation of his Romex and qualified to do a safe installation. If so, he won't have the added expense of running new wire to do a job the current Romex will handle.