None of me and things might fall
Things also can't move, as I recall
Part of me, and you are me
All of me, might determine your fate,
After all, you would be under constraint
I am a 4 letter word. Who am I?
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ROPE
None of me and things might fall
Ropes hold things up; without, they fall.
Things also can't move, as I recall
Ropes tie things down.
Part of me, and you are me
The OP is part of 'rope'
All of me, might determine your fate,
If you were a criminal, you may be hung on one.
After all, you would be under constraint
Something tied down with a rope would be constrained. Or perhaps you were 'roped' into something.
Finally:
Really Obvious Problem Everyday is an acrostic.