I want to thank everyone for their responses. The following one is particularly helpful. It sounds like they have a "reserve" price set in the system and anything over that will "win" the item (I know it's not an auction format, but...).
And actually, the "make offer" offer isn't available on all items; so perhaps the fact that it's there at all indicates that it's a slower moving item.
Originally Posted By: kozanoglu
You can offer as many times you want in blocks of 5, logging out and in to find out their minimum price they will accept. A pistol they 'sell' for $429, they accepted $340.
The offer process is instant and automatic, not human interaction.