When sending an item through 2 hoppers and then into a dropper pointed up through another dropper with a chest on top (see attached screenshot), the item will always end up in one or the other, depending on how it is made.
This will always happen one way or the other. I built about 5 different setups exactly alike, and 2 of them always had the passed item stop in the top dropper, and the other times they always ended up in the chest.
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I cannot reproduce. One item will always stay in the second dropper and the rest will go to the chest, which is expected behavior. See MC-846 for an explanation.

But it is not always staying in the second dropper. On some it will go straight through the second dropper and into the chest, and will always do it. I did see MC-846, this bug is nothing like that.

The second dropper will only send items through when it receives a redstone update because it is not being powered directly. Once the last item goes through the bottom dropper, the top one won't keep getting updates. Are you sure that nothing around the second dropper would be updating it?

All setups are exactly alike, there are no differences. I thought the rapid fire might be the issue, but it is a constant output depending on something. I have no clue what it is. If I knew how to make a movie of it I would show you, two exact builds per the screenshot, two different outputs.
My guess is that it MIGHT be related to rapid fire, but if the trigger is going to be permanent random, for lack of a better phrase, then trying to move items UP through droppers will be a pain.

Unable to reproduce either. One item is always left in the top dropper.

I just built another 5 setups. 2 in dropper, 3 in chest.

Adding another picture of 5 setups exactly alike. The redstone was NEVER active aside from each setup being used at the time.
Starting with the left setup and dropping one item into each hopper:
Setup1 from left: Item always ends up in top dropper.
Setup2 from left: Item always ends up in chest.
Setup3 from left: Item always ends up in top dropper.
Setup4 from left: Item always ends up in chest.
Setup5 from left: Item always ends up in chest.

Just made another setup FAR away from any work I have done, and: Item ended up in chest.
It is certainly random, and can easily be reproduced for me. So not sure what else to tell you other than if that happens in the production update than I can never trust a dropper to do what it is designed to do.
How else can I assist to show you this is messing up for me.

Third picture with open chest showing: Item ended up in chest, EVERY time. Bug.

After building a couple more found one which did skip the top hopper. After messing around for an hour could still not find a reason why it does that, seems random. Confirmed anyways.

Thanks for going that extra mile Tails, I appreciate it. Strange, but it does happen. It happens close to 50% of the time for me. If I can assist in any way, let me know.
Thanks!

Still happens with 13w09b as well. Just fyi

You can update affected versions yourself. Just fyi 😉

In the 13w09c still exist.

Ah, I did not know that Tails, thanks!

I think the problem came from dispensers that are too sensitive when powered. The droppers are so fast that they react at Redstone updates not like an instant update but like frequent block changes. Maybe adding 1 tick delay before item drop will fix the problem.
Please add a picture from https://mojang.atlassian.net/browse/MC-10667 It is easier to reproduce and a bit different.
Or a picture from setup like here, that should be more realible http://www.youtube.com/watch?v=E4FuwPPcpds

I have determined the cause of the seemingly random behavior of this bug. When a dispenser or dropper is powered, opening a normal chest causes that dropper or dispenser to update as if the normal chest was a trapped chest. In this case, if you open the chest when the dropper is momentarily powered when it is transporting the very last item, it will cause that dispenser to update and thus transport the last item into the chest. This is of course, not intentional and thus unless you knew what was happening, the times when you open that chest when the dispenser is unpowered would not trigger the dropper to transport the last item into the chest which is why a lot of you can not reproduce it and why it appears to be random at first glance. see https://mojang.atlassian.net/browse/MC-10997

Yuki, that can't be it because none of the chests were ever even opened.

Update Affects Versions to 10b and pre-release.
I thought the BUD nerf will do the job, but it fixed nothing. Why was the BUD ability removed then?

Still exist in 1.5.1

I am in 1.5.2 and this happened to me too. I don't know if it is exactly the same.
I have 2 droppers facing up both and placed vertically attached to each other. Sometimes the down dropper will move the item to the top dropper but not always. Even if this is not the same as your's, even though I really do think it is, I am voting for a fix.

This glitch sometimes occurs in horizontally placed droppers as well. Sometimes it will travel two blocks, sometimes one, or sometimes gets stuck...

I seem to have the same issue in 1.7.2 - though mine will generally go farther.
My dropper column seems to drive/push items through ~19-20 transfers, but no farther.
I took items out where they got stuck, and put them back a few droppers earlier, and they got stuck the same place - however! - when I inserted a new stack of material a few droppers farther along than my original 'start' and the new stack got transfered that same number of transfers and stopped at a later point than the original stack stopped.
I have dug out and verified the direction of all the droppers multiple times, as well as the redstone signal.
(oh, and I've also tried a bunch of different 'clock' speeds of redstone pulse to drive the droppers, I'm currently using a very slow one just to make sure that speed isn't causing this issue.)

I always thought that this bug was because redstone wire is unpredictable in the order that it updates blocks. This can be exemplified by having a line of droppers all being powered by the same line of redstone dust. How far an item goes down the line of droppers depends what order the redstone decides to update the droppers in. If it is intended for redstone to be slightly indeterminate, then this is not a bug, however I would argue that it is.

Dupe of MC-11193?

Is this still an issue in the most recent versions (i.e. 1.10.2, or 16w42a) of Minecraft?

Pretty sure this is a dupe of MC-11193.

Resolving as a duplicate in favor of MC-11193.