I can't speak for neural networks, but the performance of a genetic algorithm is primarily limited by the difficulty of the problem and how long you're willing to run your algorithm.
What frequently happens with standard genetic algorithms is that after a certain amount of time, the solutions tend to converge to a local optimum. Genetic algorithms work by initially creating a set amount of random solutions and then combining and mutating those solutions. The best solutions are kept and the worst are thrown away. The set of solutions is referred to as the population and one iteration of the recombination, mutation, and replacing solutions is called a generation. The user will set a limit of generations to take place before the algorithm ends.
In a standard genetic algorithm, the solutions in the population tend to become very similar to each other after a number of generations (depending on the problem type and how hard the problem is). The more difficult the problem, the sooner the population tends to converge on a single or a very small number of different solutions. There are MANY different proposed solutions for this, none of which are perfect. Researchers have developed new operators for recombination, mutation, replacing solutions, as well as entire new representations for the solutions. I think it is generally accepted that the better you tune your operators and representation to maintain good diversity within the population, your algorithm will be more successful. However, this is a VERY hard problem and there is no single solution.
The most popular conferences in this area are GECCO, WCCI/CEC, and Evo*. They'd have papers on the state of the art in the field for these things.
TL;DR - The population converges to a very small number of solutions and has a hard time breaking away from these.
I'm not sure if I answered your question sufficiently, but I hope this helps!