yes, as there would be little to no incentive for any individual to make those payments. The problem of free riders is basic economic theory.
If you wish to sue me and I have to paid to fund your court system what happens? Either I will be compelled to pay the "voluntary" fee or I will get the benefit of a court system without paying for it, so why should I pay?
If the police are effective at preventing crime, why should I pay the insurance fee? If all the murderers rapists and thieves are locked up because of the wonderful efficiency of private sector justice, what is my incentive, as an individual, to pat for the police. So long as everyone else pays I get the benefit for free.
We've had thousands of years of civilization in hundreds of different societies, why have none of them ever been able to make an-cap work?