Weights not converging while cost function has converged in neural networks
The weights in a model do not need to converge to stop training.

One possible explanation is that the model error surface has a big, wide valley. If that is the case, the loss function would be low throughout the valley but there would be many weight combinations that would all yield similar performance on the training dataset. Once a model has reached an acceptable loss function value there is no reason to continue training, just take any set of weight values.