The cool down timer of hoppers is used for picking up and pushing out items.
This leads to hoppers that push out items, to only pick up items every 8th game tick.
If items happen to cross the hopper only outside of these ticks, the hopper might never pick up items, even though there are free slots in it.

What I expected to happen was the hopper picking up items.

What actually happened was the hopper not picking up items in certain cases.

Ways to reproduce

With hoppers

(by Xavom)

there's 4 hoppers in a loop on the top, and you should first power the block then put one item in each set of looping hoppers

With hopper minecart

The minecart is driving back and forth on a 9 block long track.
In the center there is a hopper below the rail with some items in it pointing into another hopper that points back (pushing items back and forth between each other).
Depending on when the minecart started driving, the hopper will not pick up items from it.
Note that this is only the case when the hopper is pushing out items. If you break the second hopper it would pick up items fine.

The test setup is attached as structure file. mc-120596.nbt