I've just been doing this for the 5th time, 4th Horde and one time alliance and while I don't remember it being this way before, when I've had 1 key in my backpack, even though they're not marked as unique I've not had any more drop until the key is used.
I picked up a key and didn't realise, had to log out for a while, when I came back I killed 20-30 mobs with no key drop. As soon as I checked a cage and found I had a key I was able to get new key drops. I also did not get any of the big cage key drops during this time.
Having tested it further in game where I had a key and didn't use it, I continued to not get any more keys drop until I used the cage key, despite the key drop rate being around 28%.
I hope this saves you all some time.