You're using MR16 50W 12V Halogen Flood Reflector Light Bulbs is my guess. Some of these light fixture for these light bulbs have a small transformer (4 wires - 2 pri (blk & wht color) & 2 sec -- about 1 inch x 1-1/2 inch x 1/2 inch) for each light bulb. They cost as much as a light bulb (maybe more), My guess is this one was on its way out and is now open circuit.
It sounds like you have a fixture controlled by a conventional dimmer and are using CFL non-dimmable bulbs.
Standard dimmers were designed for incandescent bulbs. Many CFL and LED bulbs are non-dimmable. Some are dimmable, but only work with special dimmers designed for their electronic circuitry (conventional bulbs have no circuitry, only simple filaments). Some CFL or LED (not many) will work with a standard dimmer.
First you need to make sure the bulbs are dimmable, and then that you have a compatible dimmer.
In general, CFL and LED bulb packages indicate whether they are dimmable or not. If it does not say dimmable, it is probably not. Dimmers also indicate whether they are intended for CFL and/or LED bulbs. Again, if it doesn't mention CFL or LED, it is probably not compatible, unless you have bulbs that specifically say works with standard (or conventional) dimmers.
Check the fixture by using an incandescent bulb. If it works, either go with incandescent, find CFL or LED bulbs that work with standard dimmers or swap out the dimmer.
If the incandescent doesn't work, you have a wiring problem or a defective fixture.
Best Answer
Your type of fixture uses a Compact Fluorescent Light with replaceable tubes separate from the ballast.
Most likely, your builder fit "builder grade" (i.e. cheapie) CFL fixtures with cheap ballasts. It's not unusual for several cheapies installed together to fail one at a time within months of each other.
A fixture is made up of 4 parts: the tube, the ballast, the lamp socket, and everything else. Generally cheapies are cheap through and through, but the #1 thing to fail is ballasts, and you can obtain and swap ballasts. Aftermarket ballasts tend to be better made, top brands are GE, Philips/Advance, Osram/Sylvania.
Another option is to switch to LED fixtures and there are two ways to do that.
Get a true LED fixture which does not have removable bulbs. There's no need for LED emitters to be changeable, since they will outlive every other component including the house. Quality matters because if the onboard power supply fails, that kills the fixture. Don't buy these at big-box stores, visit 2-3 genuine lighting supply houses.
Modify the fixtures with a socket that can take a screw-in LED. However this is not to be done lightly: the fixtures are simply not built for the heat of incandescent bulbs, and can burn your house down if you cheated an Incandescent bulb in there. Unfortunately if there is a socket that can take an incandescent, some goober will put one there. So you cannot use a socket for which incandescent bulbs exist, and this limits you to things like Gu24 or Gu10.