Edit: Changed the title to make point clearer... my bad.

If you die a interesting way there should be a notification about it. But if you're an expert platformer and never died at all you should get an achievement for that but that would be the only one. The death notifications would be a mystery, too, so it could be a funny or frustrating experience when you lose in a cheap way. Here are some ideas:

Double Burn/Triple Burn: Hit twice/three times by a same enemy fireball and lost a life from it.

So close....: Killed by an enemy right before you touch a  star.

Was there every any doubt?: Killed by the same part of a level over three times.

Biting the bullet: Killed by the same bullet bill/cannon in one level over three times.

Pointless: Lose a power-up you gained in less than a second

Even more pointless: Die a second after getting a star.

....: Used a star power-up before the level started but died a few second later

Kid Icarus: Lose a flight power-up immediately after gaining it and died a few second later

That's all I got for now :P

Amazing idea!

Hahaha I'm sure xbox has this too but on PS3 these would be hidden trophies. The platformer Escape Plan on Vita actually has exactly what you're explaining.

The point is to not get them though. They're a positive punishment. It's suppose to humiliate you or make you laugh at your horrible gaming skills. These death achievements could even become actual hinderance. For example, if you get too many of these death achievements you can't access the last world so you have to replay the whole game thus adding value to the game. 

Would it not be hard to incorporate these ideas, I have never heard of a game doing this. If I am wrong, I would be gladly to accept the right answer.

This is one of the worst crimes which can be committed by achievements.

e.g., a multiplayer shooter gives an achievement for getting a hundred melee kills. Your team will really appreciate having a jerkoff running around with a knife trying to get that achievment instead of a valuable ally on the field. And is the douchebag in question having fun getting shot in the face while he tries to get 100 stabs, or he just fullfilling his completionist compulsion?

I just looked up Escape Plan and it does kind of have the achievements I want the NSMB games to have. However, (if I'm understanding the game's trophy system correctly) these trophies award the players from losing by giving them more trophy points.

I want these death achievements to humiliate or humor players' mistakes. They should be something you're embarrassed to say you have when someone asks you how many or what death achievements you've obtained. They aren't something to brag about or collect. They're something embarrasing you want to avoid. 

From here I want to digress that I'm aware of the perception of achievements making players sometime enjoy their game for collecting and achieving every reward it has rather than the acutal gameplay so I want to repeat myself that these death achievements are not suppose to act as rewards but punishments.

Well that's hard because people love discoveries. Even if the game doesn't reward you for getting them most users will want to find them all. This could be avoided by dumming down the achievements and giving them boring names but then what would be the fun of it. I'll admit it's a cool idea but to make people not want these achievements will be a tricky task.

How about this. The death punishments can prevent players acess to the secret  last world or any other unlockable feature after completing the main game. For example, getting more than 10 will prevent that acess. The player can then be rewarded for not getting these death punishments by getting notices that say, "Hey, you never died trying to reach for a star or didn't die by the same fireball that hit you twice." These rewards could be shiny trophies and the punishment can be broken old tropies to encourage players to avoid death punishments throughout the entire game. 

(Digressing again: These rewards for not getting punishments might make players  only  play the game to obtain them, but looking at the punishments I provided, it's not going to make players do random assignment like collect 50 coins from each level of world 3 to obtain 50 coins from world 3 reward. It also encourages the players to play the game as an expert, which gamers always enjoy doing. No one wants to die a second later after using a star power before they start a level or die multiple times at one spot of a level continuously.)

If you want to add "punishent" to death you could add a "challange" to unlock secret levels. You can start a challange at the beginning of each world, to win the challange you must find 4 red coins in each level and bring em to the end (of the world). You are given 5 lifes, if you die you'll lose te red coins and you'll have to start the challange again.