We can very well feel the magnetic field around a magnet, but we can't see it. Why is that so?
Also, can we cut a portion of the field and use it?
Magnetic fields are not visible because it was not important during evolution to acquire that capability.
That makes sense when you look where to find magnetic fields in nature.
There is the field of Earth, that can be used for navigation. It is used by compasses, and also some animals and even bacteria which can somehow feel the field. But it makes no sense to see it - it would look the same, more or less, maybe with lines in some direction.
Other magnetic fields are only relevant in special locations, where certain metals are found in the ground etc. and pretty uncommon.
We never had a reason to learn it during evolution, but I do think we could have learned to see magnetic fields in some way if it had been helpful.
On cutting a portion of a magnetic field:
To be exact, it's not that a magnet has a magnetic field, and another magnet has another field - there is only one magnetic field.
Not sure a black hole would count.