It actually has nothing to do with collecting another item in another FATE.

1. I collected 12 items for "A Few Arrows Short Of A Quiver".
2. I proceeded to clear the other FATEs (which were not collect FATEs).
3. A new player completed "A Few Arrows Short Of A Quiver".
4. I handed in the 12 items - without level syncing.
5. It said "You are far above the recommended level for this FATE. Your contribution will be ignored.". However, the progress towards the FATE still occurred and the items were removed from key items.
6. As expected, I got no rewards because my contribution was ignored as stated.

When I completed "Let Them Eat Cactus" and handed in half of them as level sync it was fine and I got the rewards.