I tested this in the Helgen tunnel at the start of the game. Here's what happened:
Inventory Displayed Event
======= ======== =============
48 48 Looted from dead archers.
60 60 After Ralof gave me bow and arrows.
58 58 After firing two arrows at the bear.
24 12 After dropping 58 arrows, only 12 arrow quiver on the ground.
36 24 After dropping 24 arrows, picked up 36
48 36 After dropping 36 arrows, picked up 48
60 48 After dropping 48 arrows, picked up 60
The extra 12 arrows that Ralof gave me showed up in my inventory but was not counted in the displayed number. When I dropped all arrows, suddenly all I have are a 12 arrow quiver laying on the ground. The others are gone. I couldn't find them anywhere. But I pick up the 12 and I now have 24 in inventory. It's like the original 12 stays in inventory but is not displayed. However, after that every time I dropped the arrows, they would show up on the ground in 12 arrow quivers. First one, then two, then three, etc. And every time I drop them, a new 12 arrow quiver was created.
This is a Bethesda bug. No doubt about it. I would bet it's the scripting they used to give you the arrows to start with. Maybe if you go on down the tunnel to continue the story line, they end this particular part of the quest, and things are set right again. I don't remember this ever happening before. It's certain it would be noticeable after awhile. But come to think of it, I dump iron arrows as soon as I can get steel or ancient Nord arrows. So maybe, I just never noticed this bug.
Okay, I went back into the game, loaded the save just after killing the bear when I still had the 58 arrows, and continued on down the tunnel and out into Skyrim to witness the dragon fly over. Then I dropped arrows and picked them up like this:
I dropped all 58 arrows which divided into a 46 arrow quiver and a 12 arrow quiver.
I picked up 46 and had 46 inventory and 34 displayed.
Dropped 46 which divided into 34 and 12.
Picked up 34 and had 34 inventory and 22 displayed.
Dropped 34 which divided into 22 and 12.
Picked up 22 and had 22 inventory and 10 displayed.
Dropped 22 which divided into 12 and 10.
Picked up 10 and had 10 inventory and 10 displayed.
Picked up all the 12 arrow quivers from the ground and had 58 arrows and 58 displayed.
This sequence seemed to work like the bug in reverse. There is definitely a separate tracking system for the 12 arrows given to the player. That's why there are always 12 arrows dropped separately. But these dropped 12 are not deducted from whatever is tracking them. So that when you pick up all the arrows, it just adds them back to the original 12. And you end up with 12 extra arrows each time you do this.
What I did out of the cave, by not picking up the 12 each time was to flush the total until the last 12 were dropped. At that point it cleared the other tracking system, and then counted the correct number as I picked up all the 12 arrow quivers.
So there is definitely a bug when Ralof gives you 12 arrows. And it is repeatable, not some fluke.