Is it possible to arrange three rings so that they are connected and cannot be separated without cutting one, but once any one of the rings has been cut, the remaining two can be separated without cutting either?
Borromean rings work, since none of the rings are actually linked with any other - they simply are weaved in such a way that as a group of 3, they are unseperable. Cutting any ring will let it move out of the way and allow another ring to be removed, at which point it's a trivial disentanglement.
Just in case this is a trick question, no.
You can't join 3 rings into a borromean configuration (how are you joining them together??), but you can join shapes which can then be welded into rings.