I haven't worked much with HID's but to my knowledge I would imagine that the actual HID bulbs can burn out/stock working. If they didn't there would be no reason for replacement HID individual bulbs. So since there are, I'd probably say that they do.
HID's used to be called ARC lamps since that is how the light source is generated. There is no element to burn out or fracture. The things that can go wrong are more electronic ballast and physical damage related.
The bulbs are made of glass and can be easily damaged. The glass bulbs are filled with gas and have been known to leak on some of the more poorly manufactured units. Oils from your hands can cause hot spots on the glass surface which can cause the glass tube to fracture. Make sure you don't touch the glass bulb or clean the bulb with am alcohol swab after.
The ballasts are the main source of failure for HID's. Ballasts are basically voltage converters. They convert high current/low voltage DC to a high voltage/low current source. Many of the after market HID kits out today are using poorly designed and constructed ballasts with rather high failure rates. Some don't contain proper safety circuitry in case of shortage and others will simply overheat and fail. Other problems are related to environmental damage due too poor weather resistance and subsequent moisture intrusion.
My personal recommendation is to stick with brand names on the ballast and bulbs. Sylvania being one of the more well known, and reliable, companies. Ask the supplier of the HID kit, who is the manufacturer of these components. If they don't know then go on to another source.