To be a bit more specific: There are two ways the inbox can be full:
1) 100/100 letters - simple, they just reached the maximum amount of letters they can hold and need to delete some.
2) To many items held in the inbox - you cant have more than 20 items in your inbox. So even if you havent reached the 100-letter-cap, you might have to many letters that have items attached to them. Accept some letter and transfer the items to your inventory.