Genetic algorithm neural networks converges, but suddenly stops
Well, I'm not an expert at genetic algorithms, but here's what I would check, coming from a neural networks perspective:

1) Learning rate (in your case, I think this is your mutation rate? Please correct me if I'm wrong). When a neural network exhibits this type of 'convergence', often you need to decay the learning rate either as a step function after a fixed number of iterations, or smoothly through exponential decay. Is that an option for you?

2) Initialization. Have you tried initializing your algorithm from different points and seeing if the convergence behaviour changes?

3) Adding noise. Sometimes you can get stuck in a poor local minimum, or at a saddle point. Small neural networks are more likely to get stuck in local minima. Saddle points proliferate in high-dimensional space. When learning stalls like this, adding some jitter to your parameter values and trying to learn from that new location can sometimes move you from a saddle point or local min to a better location in error space, and learning can continue from there.

4) Hyperparameter selection. How do you know your network is the correct size? Have you done some cross-validation to determine the number of hidden units you need, for example? Have you tried any kind of regularization?

Hopefully that provides some inspiration, or at least a starting point. Best of luck!