It's simply up to your own discretion. As long as you follow the 3 basic rules (only catch the first Pokemon in each area/ route, if a Pokemon faints it is dead and has to be boxed/ released, and nickname all Pokemon caught), you're allowed to add other rules to your own liking.
You can take it as a mercy point, because your buddies can be pretty useless in battle, so the double battles that you face are really hit & miss (not so much Riley and Marley, but Cheryl and Mira are super useless from my experience).
However, some people will say that it's just tough luck if Cheryl's Chansey decides to miss 5 times in a row with Egg Bomb, and your fainted Pokemon is dead regardless, so the decision is really up to you. In the latter case, you just put the Pokemon that fainted to the back of your party and don't use it, then box/ release it when you reach the next Pokemon centre.
What defeats the purpose of a nuzlocke is inconsistency, e.g. you don't care if your Budew faints/ "dies", but when your Kirlia faints/ "dies" you give it another chance because you happen to want a Gallade.
Decide on your rules from the beginning, and stick to them until the end.
