Technically, yes, you could. Though it would serve little practical purpose and be a really expensive proposition in this day and age. When you enroll in a health insurance plan there will always be a question asking if you will have other coverage in effect along with the policy you are applying for. This is so the two insurance carriers can coordinate their benefits among each other. In no case will your coverage with the combined insurance policies exceed 100% of your covered expenses and it is illegal to make a profit off of having health insurance coverage. So, you would pay two full premiums for two separate major medical policies but, at best the secondary policy would only pay what wasn't paid for by the primary policy.
While I haven't seen or heard of it being tried, I would expect that a person enrolling for coverage through HealthCare.gov, especially a person who qualifies for a premium tax subsidy, would not be allowed to enroll in more than one plan by the way the computerized enrollment system is set up.
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While I haven't seen or heard of it being tried, I would expect that a person enrolling for coverage through HealthCare.gov, especially a person who qualifies for a premium tax subsidy, would not be allowed to enroll in more than one plan by the way the computerized enrollment system is set up.