In this video, the tested person has to foresee a sword attack from their back and evade it. Are there any ways to train yourself for such a test? Can a human-being "feel" and foresee something that threatens him from behind?
To a certain degree, yes you can prepare yourself for the godan test in Bujinkan. Although you'll hear a lot of talk within Bujinkan about supernatural abilities to sense "killer intent" (saki) telepathically, humans are incapable of telepathy.
So something else is going on. What is it?
Well if you can't read minds, what else do you have? The answer is that you're picking up signals from your real senses, feeling and hearing primarily.
What you're supposed to do in the godan test is to sit in seiza posture and let your mind go blank. Move when you sense something. And someone behind you is there to do a sword attack on your head real quick.
Someone behind you moving a sword can be heard. You hear the ruffling of their clothes. You hear the way the sounds in the room change. You hear their breathing.
Someone behind you can also be felt. A sudden shift of weight from one foot to the next, or pushing down into the floor can be felt as a vibration.
Now, the purpose of meditating while doing this is to reduce the time needed to react. You are listening to your subconscious mind (your intuition). Your subconscious is picking up all these signals from all of your senses, and that giant neural network in your head is able to process it immediately, without your conscious mind knowing about it until a second later. So your conscious mind needs to just get out of the way and just let your subconscious tell you when something different is happening. And that's the goal of this exercise, to test whether you've learned how to do this when you want.
Your subconscious mind isn't perfect, though. There are many false triggers. A lot of people fail this test.
There's another way of passing: Random chance. Take this test enough times, and chances are you will pass eventually. And in Bujinkan circles, there is a lot of gossiping about how so-n-so shihan had to take the godan test a dozen times before passing, and he only did so by random chance.
As for what you can do to train this, every now and then in your regular training, let your conscious mind relax and just listen to your gut (your intuition). That doesn't mean not allowing yourself to think or letting your mind go blank. That means go with your gut. Relax more. Don't plan anything. Don't analyze. Just be in the moment listening to the feelings you get. Over time, you'll learn to trust this more as you start to have successes.
When I did this the first time in class, my Booj instructor congratulated me for taking a step into a much bigger world. Indeed, it suddenly made sense to me. It is transformational. A lot of it comes down to allowing yourself to relax and sense rather than actively thinking, analyzing, and over-analyzing.
I mean it when I say "don't plan anything". Like you might be in Bujinkan class, and someone does a stomp front kick to you when you weren't expecting it, and you react by side-stepping and grabbing it. Now you have his leg captured in your hand. What do you do with it? Well, normally you would be thinking real hard about it. You're thinking, "Oh wow, I have to do something really fast before he does something! What do I do?!" And you go through your list of favorite techniques to try. And you might try something, only for it to fail. So you try something else, and it fails, too. Then blam! Your opponent throws you to the ground hard, and you're left thinking you just suck and need to train more.
No, that's not why you failed. You failed, because you did the wrong technique (of course). But why? You relied on a list of favorite techniques and tried to match one to the situation in less than a second. Of course you failed, because you had only one second to perform this super complex analysis. The smartest person on earth won't do much better at it.
Instead of doing that, imagine you caught that kick. Now, just relax and listen to your gut. Don't plan. Don't freak out about having to do something. Whatever comes to you from nowhere, do it. Do it even if it seems wrong or stupid. You can't rely on your analytic mind when you need to make a split second decision.
When I finally "let go" (of wanting to analyze everything) and did this in class in this exact same situation, I just relaxed and stepped with my entire body into his leg. And down he went. It was so simple and obvious. And I did it without stressing out about what to do. It just happened all on its own without thinking about it.
Quite a transformational moment when you do this for the first time!
I want to add a video I saw recently of someone who described this very same thing in his own Brazilian Jiujitsu training. He said it was transformational for him as well. In his description, he said very briefly that what allowed him to suddenly make progress was when he just stopped trying to think ahead and just tried to "hold on for dear life" and wait. Then he said that things would just come to him without much thought, and he would gradually do better and better over time. It's the same thing as what I described. This just goes to show you that this is not just a Bujinkan thing or some weird, esoteric nonsense that modern martial arts don't think is useful. It's useful. In fact, I would say everyone eventually realizes the same thing, if they train long enough:
Hope that helps.