This is actually happens in the most commonly studied metric in all of General relativity, the Schwarzschild metric. It's often said to describe a black hole, which is of course true. However, this black hole actually connects two universes by a wormhole. The reason you can't get from one universe to the other is because the length of the wormhole expands too quickly, so you can't actually get to the other side. As the wormhole gets longer and thinner, eventually it gets to a radius of $0$ right in the middle, and pinches off. This is the singularity. The singularity then advances, eating up the now broken wormhole on both ends, until everything inside the black hole has gone into the singularity.

This is actually happens in the most commonly studied metric in all of general relativity, the Schwarzschild metric. It's often said to describe a black hole, which is of course true. However, this black hole actually connects two universes by a wormhole. The reason you can't get from one universe to the other is because the length of the wormhole expands too quickly, so you can't actually get to the other side. As the wormhole gets longer and thinner, eventually it gets to a radius of $0$ right in the middle, and pinches off. This is the singularity. The singularity then advances, eating up the now broken wormhole on both ends, until everything inside the black hole has gone into the singularity.