Unwinnable
While any given game of NetHack may be very difficult to complete, it is generally understood that every game is winnable — that is, no game is created that the player cannot win.
There is, however, at least one way that a game may be rendered unwinnable, mostly by accident. If the player unsuccessfully performs an altar conversion by sacrificing at a cross-aligned altar with negative alignment, he or she is converted to the alignment of the altar. Permission to proceed down cannot be obtained from the quest leader nor can you go down or level teleport from the quest home level even with prior permission if the player is not of his or her original alignment, and both are necessary to obtain the Bell of Opening from the quest nemesis.
Therefore, if the converted player is not in the quest branch below level one at the time of conversion and does not yet have the Bell, the game is generally unwinnable at this point. A player's alignment cannot be converted back to any previously held alignment by means of an altar, and the quest leader will "see through" a helm of opposite alignment.
A second case is if the quest portal has been removed because you have angered the quest leader, or have failed his alignment record test seven times. In this case, you will not be able to enter the quest branch at all by normal means, and you will have to pacify your quest leader before you can get permission to go down. Don't throw the silver bell at him!
There is a slight exception if the player is neutral (or changes his alignment to neutral in the first case only), is not a wizard, and has previously entered the quest, below level one in the first case: the player can wish for the Eye of the Aethiopica to branchport to the quest. (You have to have visited the quest once already because the Eye will not open a portal to a branch you haven't been to, and it will only drop you as deep as you have already gone before.)
Thirdly, if the quest leader is killed before you get permission to go down, the game will also be unwinnable. In practice, this does not play a role. Due to a bug, a petrified quest leader brought back with stone to flesh will not grant permission.
The best a player can do in an unwinnable situation is to escape the dungeon with the Candelabrum of Invocation, the Book of the Dead and any other sources of bragging rights.
Note that simply putting on a helm of opposite alignment does not render the game unwinnable--removing the helm (after uncursing, if necessary) restores the character's original alignment. The message you get if you try that in the quest branch is somewhat misleading.
SLASH'EM
SLASH'EM does not actually add any new ways to make the game unwinnable, although it certainly adds ways to make it nearly so. For example, although one needs two of the three the alignment keys to reach Vlad the Impaler, lacking them, it is possible to wake him up with a mind flayer's psychic blast. He will then teleport to meet the player, and be carrying with him the Candelabrum. Also, locking vital items behind the doors and destroying the keys doesn't make the game unwinnable either. One can retrieve the Amulet or any Invocation item by luring an appopriate covetous monster into the the third level of Vlad's Tower, and the monster will then retrieve the items for the player.
Sporkhack
If the quest leader dies, the stairs are automatically unlocked, in order to avoid issues with unwinnable games.
