First the Bad.
On the extreme end of purely adversarial games, is a mechanic that many people dread. Player elimination. Losing is one thing. But being eliminated?! And then everyone just plays on without you! About the most pure example of a mechanic which is fun so long as it never happens to you. And especially not if you are first. Of all the stories of rage quitting that I've heard, an early player elimination is the most common theme. Especially when the average length of the game in question is basically an entire day. Nobody wants to be kicked out of the game, with the prospect of having to wait hours, all alone, for a chance to play with people again.
Some games are awful because they were designed around player elimination concepts, but don't have player elimination. Instead of players being eliminated and the game ending, players are forced to continue on, in some sort of zombie state, until the end of the game. They get to do a single default action, over and over, every turn. Like taking 2 resources. Or placing a single combat unit in a single area of the map.
Some games commit an even worse sin of having strange and bizarre catch up mechanics. They artificially launch a player that is about to be eliminated way ahead of the others. It can be immensely frustrating to watch a player fail upward in a game. But sometimes the system just rewards that. Sometimes a player who is especially adept at min-maxing can divorce themselves from the theme entirely, and do something so counter intuition that it feels like cheating. Although not based around a catch up mechanic, the controversial starvation strategy in Stone Age comes to mind.
It's a situation I've encountered occasionally in Cyclades with a particular play group, in a three player game. One player was pushed into a single territory, and began partaking in the catchup mechanic that gives you an incredible amount of gold turn after turn so long as you are on only one territory. Me and the other player were forbidden from attacking him according to the rules. You are not allowed to eliminate a player, unless it allows you to win. Knowing this, the isolationist player never built anything that would give us a pretense to attack. Meanwhile me and the other player kept each other in a very fragile balance of power, unable to get ahead at all. At some point the isolationist player decided he had enough money to win in two or three turns and swept the board. There was nothing that could be done to stop it.
So when is play elimination appropriate? If it's a short game it isn't so bad. Bang is a good example. If it triggers the end game it isn't awful as well. At least in either of the situations, you know that even if you are being eliminated, you can jump right into another game. No need to wait forever.
Dog Eat Dog.
The biggest problem with a player elimination style games however is the cannibalizing of one players abilities by another. It's akin to having your arm ripped off, and then being beaten to death by it. The quality of one player's game experience is being enriched at the expense of another. As they gain more longer more diverse turns, the other players turn is getting simpler and shorter. The capabilities of the game are too intrinsically tied to what one player can take from another.
Some games have mitigated this in interesting ways. Cosmic Encounter, though it does not possess player elimination, has a lot of the hallmarks of a player elimination game. Players are conquering planets from other players. However, a player always has a special ability not tied to any particular planet. As well as a hand of cards independent of any of their planetary holdings. These extra-territorial powers keep a players turn enriched, despite the fact that their physical holdings are being devoured by other players. It keeps them engaged thoroughly right until the end. As well, even though it is theoretically possible for a player to lose their special power by losing too many of their home planets, it's rare since the players can only attack the player a balanced deck of cards tells them to attack. As such, it is strictly impossible for a player to lose all their planets in a row.
So having territorial bonuses, as well as extra-territorial powers, go a good way towards solving the player cannibalization problem. However, if done poorly, it can easily cause mechanic creep. Cosmic Encounter works because it has a very simple combat, diplomatic, and victory point system. And they are all tied together remarkably well. Other games attempt to accomplish this far less elegantly. A complex war system coexist with a complex economic system alongside a complex science or culture system, which all stacks on top of a complex resource system. These games can be incredibly fun and epic, and yet, it always seems like it was the person who knew the rules best that won. Note I didn't say knew the rules. Just knew them the best. Because knowing the ins and outs of the rules perfectly just is not practical in a game of that scope. Especially as often as the typical "epic" game is able to hit the table for most busy adults.
So, to actually answer the question, when is player elimination appropriate? When it ends the game. When the game is short. When a player has other resources to draw on, which cannot be taken from them. And when said resources don't overly complicate the game. Then perhaps, player elimination is appropriate, and maybe even still a pleasant experience.