Yes that happens quite often. The really good interviewers know that and understand that their job is to help the candidate to present his/her skills. That means additional questions, rephrasing the question, patience, empathy and first of all: taming the premature judgements.
It's really hard but it can and should be done, if interview is to have any sense.
If you just take their initial response and move on to the next question, you fail as an interviewer. You've got to go deeper... that's also the way that you can quickly spot a bullshitter or someone making a proverbial mountain out of a molehill project.
I think a lot of candidates think they're immediately screwed if they don't have a compelling answer to every question. It's not seen that way on the other side of the table, at least not when I'm there. Often the interviewer himself is just trying to find a foothold that he can take to really start digging into something. Good interviewers throw out a bunch of questions that are designed to get you talking until you start talking about something they can sink their teeth into. Bad interviewers have no idea what to do and throw out stupid, random questions that sound like job interview questions to them, accept your answer, and then do it again for the entirety of the 30-60 minutes. At the end of your interview, they'll make the same decision they would've made if they'd only seen your picture and were told they had to decide from that (unless you managed to offend the interviewer during your responses ;) ).
Any nervousness I had as a candidate was permanently resolved after I started interviewing people and seeing how others reacted to interviews. Most interviewers are not disqualifying you for all the stupid crap you imagine. If you're good and they pass, they're either that incompetent in which case you don't want to work for them anyway, or they have many good candidates and the choice is then based on unknowable differences in perception like perceived likeability or other superficialities.
In that situation, if someone asked "tell me what special activities you've done in the past month", I might point to that one time where I washed the dishes, whereas my wife, not seeing dish washing as anything impressive, might say she's done nothing special. Even though I only washed the dishes once, I come off looking better there since I have an example I can point out.
Simply saying "I did my job every day" isn't good enough. However, not doing your job except on one occasion makes that one occasion seem special.
What actually happened was you just sat there browsing reddit until the deadline had almost arrived, then worked like crazy trying to get the site finished. Normally your team wouldn't work contracts like that (which means normally your team would do their job properly). And it was a challenge and a test of your skills, because normally you don't do jack squat at work. In fact, the reason you're looking for a new job is because if you don't quit your current employer will fire you.
Now you have a great story to tell an interviewer about how you completely fail to do your job properly. But by dressing up the words a little bit and using some creative euphemisms, you can make it sound like you went above and beyond. And it's not really a lie, because you did go above and beyond... eventually. You just didn't have to do that if you had worked properly. All you have to do is leave out the fact that you're a massive slacker.
So you get the job because you told an awesome story of how devoted you are, and all you had to do was leave out one single fact (that the deadline was only tight because you procrastinated). Meanwhile your really awesome coworker tells every single fact 100% true and you get the job instead of him, because your story sounds better.
If the interviewer follows up with all the team members who were actually there it wouldn't work, but I kind of doubt this happens a lot.
For example, one could make the argument that they favour people with good situational recall, because they're better able to think about occasions where they've done something that applies given the criteria specified. Or that it favours people who have depression, as they tend to be more self-aware and self-critical, so have a larger bank of answers readily available.
Unless one is talking many years prior, these situations are still in the memory. The question isn't meant as an approach to rehash a particular solution. It is meant to set ground work to explore decisions, options presented, etc.