Solution 1:

I think it means how many times you downed someone but not finished. Well, check the images below out.

1-man squad, 2 DBNOs

As you can see, I played 1 man squad, and now, the final proof [picture below], I actually downed 1 guy but not finished, downed another and he died to playzone. I killed his mate and the other one standing from another team. Got 2 DBNOs. To the ones still wondering/with doubt, how would I get downed but not finished playing solo? 2 DBNOs by the way.

So it must only means how many times you downed someone but not finished.

1-man squad, 2 DBNOs

Solution 2:

This is a more general answer, as I know the term, but not how it would apply to PUBG...

DBNO stands for Down But Not Out. I believe the terminology really starting becoming a thing when Gears of War came around, but there certainly were games before it that would use this mechanic (though that is not in the scope of this question).

It basically means your character has been knocked down or out or incapacitated in some way, but are not dead or out of the game. Usually a team mate would have to come revive the player, or there are ways to bring yourself back into the game (waiting, medkits, crawling to a safe area, etc).