Can I sue the insurance marketplace?

Upon the birth of my daughter in December, we enrolled her in a plan through the marketplace. In April, the marketplace terminated her policy because they hadn't received a copy of her SS card (I sent in a copy of it in January, as soon as I got it). since we were on auto-pay, each month the premium was deducted automatically from my bank. In August, I called the insurance provider to update my credit card info, it was then when the rep told me the plan had been terminated (which I was never notified about). During the 3 months of not being insured, my daughter had a doctor visit that amounted to $500. "I" am now responsible for this bill, which I shouldn't be since I was never notified about the plan being terminated. The marketplace said they would do nothing about it. Do I have a case?