This especially happens with slimes. When I kill a big slime and then shoot in the the group of little slimes, it bounces off an invisible mob. It doesn't happen every time so it might be hard to reproduce.
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I managed to reliable reproduce this bug, uploaded screenshots.
Step 1 - create a machine like that. It's a clock linked to a dispenser. The machine is high enought that the zombies will die of fall damage.
Step 2 - Fill the dispenser with zombie eggs
Step 3 - Start the machine (lever starts the clock)
Step 4 - Shoot the zombies as soon as they die. Use a flame bow to make easier to see the arrows bouncing.
Sorry for the lag. You can kind of see the arrows bouncing off air in this video though.
Related to MC-1246
This one still occurs for me in 1.4.6 in both single- and multi-player, though more often in multi. Possibly due to a client/server communication problem regarding arrows' current coordinates? Will try to also get video; it's very bizarre.
Confirmed on the two last snapshot (at least) (One time, i saw my arrow turn back during its fly)

Is this still a concern in the current Minecraft version 1.8.1 Prerelease 3 / Launcher version 1.5.3 or later? If so, please update the affected versions in order to best aid Mojang ensuring bugs are still valid in the latest releases/pre-releases.
Resolved as far as I can tell.

Resolving it as such, thanks.
If someone else can still reproduce this, please leave a comment and request a reopen.
I produced this "resolved" bug and attached a video of the bug, I was running the 1.8.1 release. If a mob, in my case a zombie, gets moved in one direction and you shoot arrows at him from a 90° angle the arrow will bounce quite frequently off him, with different angles it happens less often. I teleport the zombies by 0.3 every gametick with:/tp @e[type=Zombie] ~ ~ ~0.3. The zombies get spawned in with NoAI: /summon Zombie ~ 40 ~ {NoAI:1,PersistanceRequired:1,Silent:1,Attributes:[{Name:generic.maxHealth,Base:1}]}.
So this Bug is still around and it is quite annoying when you try building a shooting range with moving Zombies.

@Joel Henning: Invalid since you are using unsupported modifications.
So I rerecreated the video with normal Minecraft 1.8.1 no Mods, just normal Mineccraft. And there you go it the arrows still bounce off the Zombies. I also tested Skeletons and Pigmen, same result, about 50% of the arrows bounce back.
The rest of the build is the same as last time, the Zombies get teleported every gametick by 0.3.
Its still invalid as you are using NBT tags, as well as teleporting the zombies every gametick. Does it occur without the teleportation and without the use of NBT tags?

I don't know if this is the same bug, but it happens to me in 1.9er versions and 16w21a. It seems to me like it was new when shields were introduced and for some reason the arrows sometimes bounce off of entities even though they don't have shields (e.g. just happened to me with a bat). Is this still the same bug or a new one?
Probably a separate issue.
Bug is back in 1.13.2

Confirmed. It seems to be much more rare than before.
I have this kind of problem with other mobs, mostly skeletons. Sometimes I shoot them and my arrow bounces and hit me.
I know that it's my arrow that hit me and not the skeleton because I have a Flame I bow and the skeleton have a regular bow. Looks like if I shoot when the mob is about to die, my arrow bounces. Also happens sometimes when I try to hit a cave spider behind a fence.