I used to do a lot of interviewing, for competency based interviews. (As Darren has said, they tend to be: "Give me an example of a time you..."). In many cases the panel will have agreed before hand how long they intend to spend on each area eg analytical skills, managing conflict, diversity etc. For example, that might be 10 minutes per competence. It doesn’t take long for 10 minutes to pass and many candidates waste time with information that is not relevant to the question. They spend precious minutes talking about strategic changes to the organisation they worked for or issues that were outside their control. What the interviewer wants to hear is a quick focused summary of the problem or task you faced, and then what you personally did to sort it out, explaining any snags you overcame, and what the final outcome was.
If you give a good powerful example in a short period of time, you can then say to the panel, “I have another example if you like.” If they say yes, then you get two bites of the cherry and may score higher if both are good examples.
So have spare examples for every area they may ask you about. (You should know what those areas are from the job advert and application form).
Don’t be afraid of admitting to having had a problem or two with a difficult task. I was always suspicious of people who claimed to have undertaken a difficult project that had no snags, came in on time, within budget, and to the right quality. Real life isn’t like that and they are usually either lying or it wasn’t a very difficult task. If you had a problem, explain what it was and how you overcame it. Perhaps you had to go back to the person who tasked you and ask for a bit more time, more money, technical help or a spec change. Or, for example, a customer or colleague who is really angry and aggressive doesn’t always calm down straight away when you tackle them. Sometimes you need to be persistent and assertive or indicate you intend to call assistance etc. Only then do they alter their behaviour. As long as you eventually manage the situation effectively, that’s fine. Being honest about snags and setbacks adds credibility to your examples.