I found out why broken rings are hard to pick up at any depth, and why chains are hard to pick up, also.
Someone else researched this, not me.
"When using a detector, and this varies greatly between what methods your detector uses, it creates a magnetic field that shoots into the ground. As you pass over a metal object, let's say a coin for now, it creates an electric current in that item. This current is called an Eddy Current. This current moves throughout the coin and creates its own magnetic field. This field is what the detector coil sees and interprets. It then takes all the input and either beeps a certain way, shows on a screen or both.
Now to the matter of different size, shape and wadding of items. These Eddy Currents flow differently depending on the items conductivity. Coins, nails, pull tabs, etc, have varied levels of conductivity and the detector usually reads them as what they are, but, if you change the shape of these items, say if a silver ring is not a "full" ring, or a is an unclasped silver necklace, or a peice of foil is wadded tightly, it will change the conductivity of that object. Not the conductivity of the metal mind you, just the way the current travels through the actual item.
To put it simply, lets say you have a full silver ring (a full O shape) and a silver ring that does not connect fully (basically a C shape), when the electric current passes through the full ring, it is able to travel in all directions easily, versus the incomplete ring, where it loses some current out of the broken ends."
And chains...
The magnetic field generated by the Eddy currents induced by the transmit coil on the targets are very small in the case of a chain. Basically, because of the contact resistance between each link is relatively high, the Eddy currents are each confined to a single link in the chain. So, each small link in effect becomes a target. Because each link is small, it can't generate a large field for the receive coil to detect. Also, the links are pointing in somewhat different directions, so their individual magnetic fields don't readily add up to a larger field which would make for easier detection.
For the above reasons, chains are hard to detect. What usually "give them away" is an attached medal, or a sturdily built clasp.