Well, when a statement is ended with "Now, who believes all this was real?", I am inclined to dismiss anything previously stated and move on with my day.
It shifts quickly from a recollection of events to a storytelling when the wording is placed the way you have yours.
As far as actual factual precognition goes, I don't think it is reality.
At least not in the sense that the writers of the world portray it.
Precognitive experiences fall more in the science fiction realm for me I am afraid.
From what I have gathered about the way time works I would say that we are capable of appearing to see things before they happen at times, when in reality we are just experiencing a momentary jump in our perception of the world.
We are perpetually living in the past, in a sense.
Say a ball is thrown at you and you react by reaching out with your arm and catching it.
The moment the ball is flung at you there is a time delay between when the event occurs and when your brain recognises that it is moving toward you.
That very tiny time delay between an event occurring, light bouncing off the event then back to your eyes, then that information being processed through your optic nerves in to your brain, then your brain making the decision to do something about the event, then the electric impulse being sent back through your nervous system and making your limbs/facial features/heart rate perform an action suggests that we are in fact existing a fraction of a second in the past. An event is already ahead of where we perceive it to be before we react in any way shape or form. So I think that when someone performs an amazing feat of reflexes, or dodges a punch almost before it is thrown, or has the apparently precognitive instinct to halt before walking in front of a bus, they are just experiencing a momentary jump forward in perception to the time in which the even is actually occurring, not when the rest of us perceive it to be.
Seeing days or months ahead, or witnessing events before they occur don't pan out for me.
If people had the ability to preconceive such things then there would be more proof of it, especially in today's world of seemingly impossible connectivity.
There would be blogs or forum threads warning of disasters, or predicting who will become president next and what their speech will include.
It is far more likely that people who think they have seen future events before they happened/met people that they dreamed about before ever meeting (as in your example) are just assigning meaning to past events in order to intensify the significance of present events.
I do not think that is an intentional choice, but a subconscious reflex that we have little control over.
We have a million little notions and predictive thoughts every day about every little thing and think nothing of it when they don't come to fruition.
But as soon as something happens and we remember being suspicious that that event may have panned out something like it just did, our lizard brain thinks "Holy shit! I totally saw this before it happened! I must have somehow glimpsed the future!"
Dreams are also a terribly unreliable source to draw from in recollection.
A vast majority of people forget what their dreams were even about within a minute of waking up, which sort of makes the potential for recalling the subject, let alone the specifics (such as what a person looks like) seem very far fetched.
Which lends to what I said before about present events being assigned significance by slightly supportive notions from the past.
An event happens and you experience some deja vu, so your brain reels back to try and figure out if a similar event has already been experienced and picks up on a similar feeling or notion you had in a dream or at some other time when confronted with some other event and overlaps them in your mind.
So I do not believe that we have the ability to see the future, but we do have a great many kinks in our perception of the world around us that our brain is constantly trying to rationalise and assign understanding to. Constantly playing catch up.
That being said, I do actually have an instance of my own where I seemingly dreamed a future event.
A handful of years ago my great-grandma died due to a few massive strokes within a few weeks.
After the first stroke my family stayed at her house to look after the place and be in the city where she was hospitalised.
After a few weeks she had a second stroke in the hospital and was essentially fading away.
My parents went to the hospital for a visit the night she died and left me and my sisters at the house.
Around one in the morning I woke up out of nowhere, already sitting up on the couch I was sleeping on and for some reason felt panicked.
While I was trying to fall back asleep I had bits and flecks of the dream come back to me and it involved my grandma in a hospital bed with my mom and dad beside her, then she disappeared from the bed like a mirage and that was it.
When my parents came home in the morning I half woke up as they opened the door (because I always, always wake up when someone enters the room I am sleeping in).
My mom walked over to the couch and said "Hey, grandma Fed died last night at two in the morning.", and I mumbled "Yeah, I know." and went back to sleep.
The night she died was the only night my dad ever went in to her room, he refused to believe she wouldn't recover and didn't want to remember her in her current state.
That, is more than enough to convince most people that at the very least some weird shit went down.
Though I do not believe I saw the future, merely experienced a strange series of coincidences that line of very nicely to suggest I had.