Not necessarily. Just as you could take a chain and stack the links such that the holes lined up, with enough precision, and a strong enough baked good, you could cut the bagel such that there were a chain of linked rings.
Well, if you cut the bagel according to this particular linked torodial method, to produce two links, you’d then be able to perform the traditional bagel slice on one of the linked halves, and that would produce a chain with three links...
But deriving from a single original torus, a chain with an arbitrary or infinite number of links, whereby each individual link is bound to no more than two other links? I’ll have to think about that one...