Would that be a bad thing to mention?
If you mention it just like that then surely it won't do any good. I suggest you mention about that subject only if prompted about it, otherwise there is no reason for you to disclose such detail.
If you feel that it is highly likely that you will be asked about that, then I suggest you give it some thought and have an answer ready in case they do ask.
How you phrase it is up to your taste, but you would do better by not denying what happened, and instead focus on the steps and solutions you took in response to such situation, and how that changed you to a better professional.
have been contemplating telling that story in the interview if they were to ask me about my biggest failure or something.
(...or something?) Probably you are over-thinking this one, there are millions of questions they may ask you during an interview, and this one is not one of the most likely.
If this question were asked then it is a tricky one. One way of graciously answering it that I can think of is to say something like: "Well, I am not certain what my 'biggest' mistake may be. However, whenever I make one I try to learn the best I can from it so I don't make it again."
This way you are not openly admitting some flaws (that we all have) and are instead proposing a constructive solution to the situation. Again, maybe you are over-thinking this a bit, but if you really feel this may happen try thinking your answer thoroughly as suggested before.