Yes, I believe that people have some ability to sense danger. It's what makes us suddenly lock our car doors when sitting in a parked car, or walking quickly through a dark, unfamiliar area. I think it can make us instinctively take a new route home from work, or slow down before we actually see an obstruction when driving. It's usually hard to prove that our sense of danger was justified, but I do think it helps keep us safe.
I also think that we sometimes get a premonition or dream of danger to a close friend or family member. While our premonition might not always come true exactly, there are times when we just know that someone close to us is in trouble.
I can't say for sure that I've saved either myself or someone else from harm, but I do believe in paying attention to intuition.