Characters are often brought to 0 hit points, by mundane attacks or magic (with some effects specifically reducing the target to 0 hit points). In following rounds, the character must make death saves, unless stabilized. A character could die of massive damage (exceeds hit point maximum + current hit points) suffered during a single effect, but in my games that is rare (would only happen in a poorly planned encounter with a low-level party).
The Periapt of Wound Closure stabilizes you "whenever you are dying at the start of your turn." Once stabilized, you begin dying again (and get a death save failure) if you take any damage, but unless you take damage equal to or greater than your hit point maximum, the periapt will stabilize you again at the beginning of your next turn, and the death saves reset to zero. So it seems the only way to kill a character wearing the periapt is:
- Do massive damage.
- Damage the 0 hp character with two or three separate attacks/effects in one round.
- Remove the periapt.
As I mentioned above, 1 will be rare in my campaigns. 2 and 3 seem unlikely during combat, and I would only have it happen if the party was facing a long-time adversary who was aware of the periapt. "You again! Minions, attack Fred, and don't stop until there's no pulse!"
Having a party member exit the combat due to unconciousness can be a big problem, but of course one of the biggest problems of 0 hp is that it puts a timer on the party to try to stabilize/revive the character before death. At first I thought the periapt was a neat magic item, now it seems much more powerful than I realized. Am I presenting its function correctly, or is there something in the rules that I am missing?
 As pointed out in a comment below, a successful attack from within five feet against an Unconscious character is automatically a critical hit, causing two death save failures. So three death saves in one round could come from three damage sources that don't meet this criteria (ranged, AoE spells, reach weapons, etc.), or one that does combined with additional damage from any source.