Aside from having the cash to pay them, just handling medical bills can be a problem. There's a Quicken program called Medical Expense Manager that handles bills for each family member, shows what you and your insurance company has paid or needs to pay, and tracks what needs to happen next such as pending payments or disputes and who's responsible. It also helps record spending, tax deductible expenses, mileage, and more. Too bad it's a standalone product and doesn't integrate with their other software.
This program is every bit worth the $50. Sometimes discounts are available online. Here's the link:
http://quicken.intuit.com/healthcare-management/medical-expe