Postscript ...
At least in New Jersey there are two parts to the medicaid coverage each child receives. One part is emergency care, urgent care, and primary care from a pediatrician. The other part seems to be ... everything else, which is managed by an HMO in New Jersey.
Basically, it seems to be the case in New Jersey that coverage for emergency room visits, perhaps some urgent care, and perhaps some primary care from a pediatrician can receive backdated coverage. However, all other portions of health care insurance are handled through an HMO, such as Blue Cross Blue Shield. No backdating of coverage happens for any part of a child's heath insurance covered through an HMO, at least in New Jersey and perhaps in some other states. In fact, in New Jersey coverage can only begin at the beginning of the next month after a child's application has been approved for anything other than a very narrow (although, to be fair, potentially ruinously expensive) range of health care services, such as emergency room services.
A KEY POINT TO REMEMBER, AND THE REASON FOR THIS REPLY:
If you ask a NJ medicaid customer service representative "Is health insurance coverage backdated to the beginning of the month in which a child applies?", then my experience was that they will answer by saying "yes", with no qualifications mentioned or exceptions noted. None, zip, zero. I asked this question several times and always got a simple "yes" as the answer. Also, materials that we found about HMOs on the NJ medicaid website focused on how to apply and who is eligible. The "backdate coverage or not" question didn't seem to be addressed anywhere in their discussion about HMOs, or at least nowhere obvious.
Nonetheless, any bills from health care services that would have been coordinated through the HMO will not receive backdated coverage, regardless of how poorly that fact is communicated by those charged with looking after medicaid in New Jersey.