There are only two ways a cock ring can malfunction that I can think of - and these situations are more user error than product malfunction. First, it could be too loose or in some cases too round and either not be strong enough to constrict any blood flow at all or roll off (We've had the rolling problem with the ring-o's from screaming o - they're perfectly round). It could also malfunction by being too tight, definitely the worse option, and do permanent damage to the blood vessels or the tissue in the penis. I suppose getting stuck on (a higher risk with metal or stone if they're not measured properly) could also be a malfunction and can also be very dangerous and cause permanent damage.
Can they break? Yup, happens pretty often with cheaper jelly ones. Silicone, leather, and obviously metal don't have this problem that I've seen but the cheaper jelly ones will snap sometimes when stretched too far and sometimes after they've just been used a few too many times for the quality of the material. I've seen this happen once and it didn't seem to hurt him at all, the ring just popped apart and fell off.