Usually, when a person says that, they have a great attachment to their self-pity and they don't actually want to give it up. Usually the hindrance of doubt will come up to try to prevent them from doing such a reflection. They may not have even tried to do it but because there is such a strong attachment to their own suffering they will bring up any excuse to stop them thinking in this way. At least that's what I've seen in some people. So, if there is an unwillingness to open to the universality of Dukkha then there is not much we can do at these times. So, part of it is, yes, they are missing the understanding that Dukkha does exist but usually it is because they want to keep attached to their particular suffering, and find it difficult to open out to others in this way. That is probably why the Buddha was so skillful with his techniques in doing this type of universalizing.
One of the most famous Buddhist stories is the Mustard Seed Story. A woman called Kisagotami lost a son when he was very young, and went a bit crazy, racing around the village trying to find a doctor to heal the child. Until finally someone told her to go to see the Buddha. The Buddha could have said to her, "Well there are other people who lose children also. Try to think of those people." But at that particular time Kisagotami was suffering a lot from her loss, and so Kisagotami asked the Buddha, "Please heal my child." But the Buddha said to her, "I will heal thy affliction." As she didn't understand what he meant, she was very happy that he would heal her affliction, heal her child is what she thought. So, she was very happy and the Buddha said, "But I need a mustard seed."
Then she was even happier because a mustard seed was a very common ingredient in Indian curry. So, she started to race off to the village to get the mustard seed, but he said "Wait, wait, I need a mustard seed from a house where no-one has died." Well, in Indian society, people used to live in extended families under the one roof, and she didn't understand what he had meant. So, she raced off to the village looking for this mustard seed to heal her child. She went from house to house and they were quite willing to give her a mustard seed but then she said, "Has anyone in this house died?" and they would say, "What are you asking? Well my servant died yesterday, or my grandmother, or my brother." And so she went to every house in the village and she couldn't find a mustard seed from any house where no-one had died. She started to understand this: that death and loss are universal. She started to understand this by experience, and then she realized that her child was dead.
She returned to the Buddha, and he asked her, "Did you find the mustard seed, Kisagotami?" And she said, " I couldn't find a mustard seed from any house where no-one has died." She opened out to the universality of death and suffering, and became quite ripe for his teaching. Apparently he gave her a teaching on impermanence and she became a stream-enterer, first stage of enlightenment, at that time. So, the Buddha realized, perhaps, that she wasn't ready to hear just a discourse about it or a bit of advice about it. She had to go through a little bit of experience on her own. And he used a very skillful means to get her to open out to it.
So, sometimes people need to experience these things and can't take our advice, and can't open out to the universality of suffering because they are still too attached to their own grief, at that particular point, or their own self-pity. And usually there is not much we can do until they are ready, unless we happen to be as perceptive as the Buddha and know exactly the technique that is going to break through to them. But that is pretty difficult to know.