Research using positron emission sleep and spend longer periods awake in tomography (PET) to measure brain between dreams. In general, you will only activity has found that people who remember a dream if you wake up during tend to remember their dreams in or immediately after it finishes.
And even the morning have more activity in an if you do wake in the middle of the night, area of the brain that processes many dreams fade by the morning external stimuli. These people are more because they are too jumbled to make an likely to respond to sounds while they easily remembered sequence.