A lot of us seemed to have lost one bird cage while doing this quest and there is no way to finish it.
Could you just add the option to get, or even buy, a new cage from the quest giver?
My situation is the map shows that I haven’t picked up the last cage, but my inventory shows 3 completed cages. Talking to him doesn’t complete the quest however.
Same here. They should put a NPC in the game that permits you to reset a quest/side quest if it is known to be bugged.
i know this sounds bad but try reloading a savegame just before the quest or simply the savegame when the quest starts and play it again
(had to do it myself 2 times)
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A bird in hand was one of the first side quest that I started. Now after 150 hour it’s a little difficult to restart from a previous savegame. 
Fortunately I was able to obtain the Completionist achievement even with some broken or uncompleted quests.
well thats a worst case scenario
On my try at this quest, I noticed that the wrong map marker disappeared when I picked up a cage. I reloaded, and before I picked up a cage, placed a custom map marker over the quest marker that disappeared. Picked up cage A, marker for cage B disappeared. Walked to my custom marker, there was cage B. Grabbed that one, and the bugged marker A disappeared. Went and got cage C, then turned in with no problem. All the cages were there, it was just the quest marker that vanished.
serious question, not trolling, but how are so many people “losing” cages? I’m on xbox one and have beaten this quest twice no problem…first time I dropped a cage the marker didn’t pop up but I paid attention to where I put it because I never knew for sure if a marker would tell me anyway so I just found my cage, re placed it and it worked…what’s happening to everyone else?
I see some people are having markers not disappear but what about “losing” cages
After capturing successfully 2 birds I put the third cage (the norther one) in the right spot, then skip an hour. The bird is recognized as trapped (in fact you can’t hear him singing), there’s a mark identifying the cage on the map but the cage is nowhere to be seen.
That’s weak. Have you gone to the spot where it is supposed to be and waited? Not sure why this would work but worth a shot I suppose
or, if it’s a possibility with your saves, load one before the mission, get to the Hunstman and save right before you start the quest again so if you have to do this again it will be easier on you, but try laying the cages and picking them up in different orders maybe