We all experience perceived fails in our career. This is nothing to be embarrassed about so do not hesitate to describe a failure you've experienced in the past. The key to a great answer is to include how you learned from that failure, and how you improved your work performance after learning from the failure.
Try to keep your example non-critical; for instance, avoid telling the interviewer how your team forgot to order new inventory, and it cost your company $15,000 in lost sales. Avoid blaming any particular person and instead, use words like 'we' and 'team' to describe the situation. Finally, be sure to tell the interviewer that, while you were very disappointed in the group's failure, you took the opportunity to learn from the experience.
"Our team was recently on a project with a very tight deadline. We knew the client and their needs very well which I believe made us go into the project overconfident. Because we underestimated the work involved, we missed our deadline by 3 business days. Thankfully we had a great relationship with that particular client so we were able to recover from the missed deadline. Although it was late, we worked overtime and delivered an exceptional project, in the end. This experience taught me to never underestimate a project and the potential roadblocks. It's always better to under-promise and over-deliver, rather than the other way around."
"In my current position our team was given a new client who needed us to recruit 5 employees. They were very specialized roles and the client was not open to candidates who needed relocation assistance. Our competitor ended up finding the right candidates before we could. It was really disappointing and I believe that we failed because we were looking too much at the roadblocks rather than thinking creatively. We learned from our mistake and our team was more aggressive with our next project."