I've been in so many situations like that its hard for me to compile what i've learned that may be helpful to you. Lets see... First of all, if you can't do a task, identify why and identify what you need to enable you to do it. It may be that your other responsibilities take precedence and there's nothing you or "they" can do to change it, but if you say " Hey, these are my responsibilities, when those are taken care of, here's the time I have left for this project. In order to complete the project I need more time." or perhaps its not time but some other resource. Like a helper or some documentation or a contact that you can call for support. Sometimes letting a superior handle gathering the resources while you work helps a ton. Or if some of your responsibilities can be pushed on someone else, your superior might make that happen if your project is important enough. But to accomplish that your superiors need to know the truth and the factors for your success.

You may not want to say to them "i cant do X, i thought i could, sorry". Instead say "to complete X I need the following things to happen" and let them decide how to get those things for you. One of those things may be pushing the deadline out to give you more time. Approach them with confidence and you may all come out ahead. If given all the factors they decide you can't complete the task, they may hand it off to someone else and at least you got the initialization done right? That's a start. And in that event did you fail? No, you did what you could with your resources, you truthfully made your superiors aware of the situation, and they took whatever steps they needed to manage the project. I don't see that as a failure at all. I see that as great experience, a beginning for the project, a step towards taking responsibility for yourself, and lastly and my favorite.. you got out of a situation on a positive note rather than negative. Saying you "can't" is negative. You can, just not with your other responsibilities.