The bug

Using /replaceitem you cannot set the horse saddle to air (remove it), one would expect it to function the same as any other slot.

How to reproduce

1. Summon a horse:

   /summon horse ~ ~ ~ {Tame:1}
   

2. Add a saddle:

   /replaceitem entity @e[type=horse,limit=1] horse.saddle saddle
   

   → See "Replaced a slot on Horse with [Saddle]"

3. Try to remove the saddle:

   /replaceitem entity @e[type=horse,limit=1] horse.saddle air
   

   → See "Could not put [Air] in slot 400"

Code analysis

The following is based on a decompiled version of Minecraft using MCP for 1.11.
The replaceItemInInventory() method in class net.minecraft.entity.passive.AbstractHorse requires the item to be a saddle at all times:

if (i == 0 && itemStackIn.getItem() != Items.SADDLE)
{
    return false;
}

This is unlike other mob equipment such as ArmorItems, where the contents can be deleted by replacing with "air".