I can't answer for certain given only the information that's there. To answer fully I need to know what is in the computer, what else is connected to its power supply (besides the computer and the lights), and how much current your bulbs draw. But I can say a few things without all that:
The gauge of wire you use will not change the total power your lights consume, neither will how many are on each loop. So, if you have enough lights to burn out a power supply, the wire you use or how you loop them won't change that. However, I misread your first post and thought you had five or six lights... you should definitely not use 22 gauge wire if it's going to carry 2.4 amps, which is what it would with all those lights in parallel. You can use thicker wire, or you can put only three lights on each loop, and have five loops of wire.
If you are using incandescent lamps like mine from the illuminated pushbuttons I got from Happ (# C161) they consume 160 mA when plugged into a 12v power supply. 12volts * 0.16amps is 1.92 watts each, and with 15 of them that's about 30 watts.
I have an old 300 watt AT power supply I use for testing circuits. On one side of it is a sticker that tells how much power each connector can supply. For mine it's 168 watts for the 12v connector. That's about half of the power my unit can supply. Lacking any real specs to look at right now, I'd venture to assume that holds for other power supplies as well (makes sense - two sources, 5v and 12v, half the power each...).
So, take the minimum wattage power supply your particular pc should be able to work in, compare to the power supply you have, and half of the difference is the room you have for powering other stuff off of the 12v connector.
I should also mention that you can power your lights off of any power supply you stick in there, so you can easily buy a 30-40 watt 12v power supply from RadioShack (shudder) or maybe some other parts supply store like
www.newark.com or
www.digikey.com, and those often come in a package that's easy to just bolt or screw to a wood surface, plug in, and connect to your lights. Then you have to find a way to turn the lights on and off with the computer, but there are numerous methods for that.
Or you can buy a 450 or 500 watt power supply for your PC, and have plenty of extra power.