If people have joint custody of a minor, will the minor be able to have health insurance in both states?

I'm living with my uncle, but I spend summers with my mom. There's a mess up with the paperwork, so my aunt doesn't have custody of me and I can't get health insurance for this state. Perfectly timed, I currently have a wisdom tooth growing in, and it hurts like a bitch. If the paperwork goes through while I'm with my mom, could I get health insurance there so that I could get this tooth removed, or can I only have it for one place?
If people have joint custody of a minor, will the minor be able to have health insurance in both states?
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