You dream only during the last third of a sleep cycle (each cycle last about 1h30). If you wake up while you're dreaming and then concentrate right away about remembering the dream you just had, then you will remember those dreams.
I you wake up during the other two thirds of your sleep cycle, the parts where you do not dream, then you won't remember any dreams.
If you want to remember your dreams, you have to wake up naturally, without an alarm clock and being awake by an outside noise or any other disturbance. You wake up naturally at the end of your last cycle, which is when you just dreamt. Then you concentrate on your dream and the more you think about it, the more you remember.
Dreams are not meant to be remembered by the mind. You need to consciously concentrate to remember them.
People who never remember their dreams are just people who do not concentrate when waking up while they were dreaming or most likely, they wake up with the radio and miscalculated their sleep cycles. Like if you put the alarm clock to wake you up in 7 or 8 hours, you will likely wake up in the wrong part of the sleep cycle. They should sleep 6 or 7h30 (any multiple of 1h30), not 7 or 8. Also somebody being awaken after 8 hours by the alarm will be more sleepy than somebody being awaken after 6 or 7h30 hours, because you feel much more rested if you just finished a cycle.