When I was a kid and had nightmares often, I would wake up in the middle of the night in a fit of fear. I had to make sure I would stay awake just long enough that I wouldn't go back into the same nightmare again when I went back to sleep. It wasn't just possible, it was common
I feel the same, although it does feel different. It's almost as if I have more control because I know what's going on now. It's like the stage in between sleep and being awake.
Also, I have found I have the wildest dreams after waking up halfway through the night then going back to bed.
I've only had that happen when the being awake state was barely awake. I've not had it happen if I'm fully awake, try to go back to sleep, and want to resume the dream.
I remember being able to go back to sleep and resuming my dream. It usually happened if I reeeeeeeally wanted that to happen and kept thinking about the last part of the dream while going back to sleep.
I remember when I was like 13, I dreamt about a girl with white hair. Somehow I felt love with her in that dream, pretty wild. Anyways, we were just like hanging around and walking. When I woke up my "resuming" trick didn't work because I wasn't sleepy. I still remember her and it was a dream I had more than 15 years ago.
I had recurrent dreams, like dreams that I knew I had already dreamt, but the white haired girl never repeated.
I have been having the same dream of hiking to a certain cabin, for years. When I'm awake I can perfectly visualize how to get there, and the landscape is completely fiction. I could even draw a detailed map.
The only dreams I usually remember more than a few seconds after waking up range from unpleasant to straight-up nightmares, so I'm cool with never continuing those.
To answer your question, dreams occur as the brain duplicates memory for long term preservation. Your dreams are a series of loosely connected concepts with no real meaning or overarcing plots. So, given that the events of the dream have no purpose and are a biproduct of biological functions, there is no requirement for them to continue where they left off, and furthermore the part where they left off can be completely unrelated to the parts that would naturally come next.