Why the d4?
There are a few reasons that this system specifically runs on the d4. First, it’s an abandoned die. Lonely. Nobody really likes the d4. It exists, and is brought out reluctantly. It is the ugly sweater from a relative of dice. That’s why there is a very real opportunity here to transform it, to make it something you associate not only with irritation and reluctance but fear and fun in equal measure. You should groan in the same breath that you giggle, knowing what’s in store.
Secondly, it’s symbolic. 4 is a number associated with mathematical perfection of shapes… but is also associated with death. That is why using 4 d4s is the ultimate expression of a Bad Time. Think of it as a summoning ritual, the invoking of death and misfortune as channeled through the roll of the dice. After all, the minimum you can roll on 4 d4s is a 4. And when you hit that great and terrible 16, it’s your 4th 4. Death death death death. That last sentence made you shiver, roll your eyes, or laugh. None of these is a bad reaction.
In short - it is not a Bad Time Game if you are not using a d4 in your design. End of story.