Yes you can put recessed lighting in a bedroom.
Recessed lighting can be installed in any room of the house you want it installed in.
In all my bedrooms I have 4 recessed lights and also my ceiling fan with lights and when I just need a little light I turn on the ceiling fan lights.
But when I need more light like on a cloudy day when no sun is shining or at night I turn on both the ceiling fan lights and the recessed lights.
Before I added the recessed lights to the bedrooms they were very dim and it was hard on my eyes to use my laptop or read a book etc.
I think recessed lights in almost any room also makes the room look much better as well and recessed lighting can add value to your home.
I have recessed lighting in my kitchen and even bathroom and they are worth having.
My home office also had recessed lighting which I turn on when I'm working in their on the desktop computer as it makes it easier to see and reduces my eye strain.