(modifying a clause or predicate) Perhaps; indicates that the proposition may be true (is not certainly false) regardless of any facts or circumstances known to, stated by or implied by the speaker.
It was possibly the costliest mistake in the organization's history.
Possibly, they will make gains in the midterm elections.
This rare and possibly unique specimen must be conserved.
(modifying a verb) In the realm of possibility; indicates that the action may successfully be performed (is not impossible) regardless of any facts or circumstances known to, stated by or implied by the speaker that might limit the performance.
I couldn't possibly be there on time.
I couldn't possibly cheat on my wife.
I'm much stronger than you, so you can't possibly win.
It seems unlikely, but, yes, they could possibly win even now.
The police don't know him, and the information they have leads them to think he could possibly have murdered his wife.