The standard thing that happens is that the wormhole pinches off and becomes singular before you are able to emerge out the other side. The maximally extended Schwarzschild black hole is a typical example of this. This solution describes two asymptotically flat regions of space connected by an Einstein-Rosen bridge, i.e. a wormhole. If someone were to try to traverse the wormhole, they would find that after jumping into it, the wormhole begins to stretch along the length, while at the same time it shrinks in diameter. This stretching/shrinking happens fast enough that it is impossible to reach the other end: the radius goes to zero before you reach the other side, and you run into the black hole singularity.
Although the exact Schwarzschild solution is an idealization, there is a large body of work that argues this kind of nontranversability is generic, provided that the matter satisfies certain energy conditions. Some discussion of which energy conditions are violated if the wormhole is traversable is given https://arxiv.org/abs/gr-qc/0405103, along with the references cited in that paper.