I don't claim to be an insurance attorney, but I did take courses in 2 different universities on US and European insurance law. One of the basics of insurance is this: you cannot be paid twice for the same cover. (Slight simplification because there are stated value policies, and there are also situations where two parties each with an insurable interest get paid in full by their own insurer. But this is not relevant to health insurance.) One policy may be primary and the other secondary; a travel insurance policy for example will be secondary to any valid and collectible health insurance you may have. Ditto for, say, "free" credit card (Amex, for example) cover.
But if you have two policies that are intended to be primary -- and that would include, say, a Swiss medical payments policy and a US policy such as one written under FEHBA (Federal Employees' Health Benefits Act) -- you will be asked to state, when claiming benefits, whether you have any other coverage. This might be workers' compensation, Medicare, a foreign or domestic policy in your name or that of your spouse or parent, and so on.
If and only if there is double coverage, the two insurers work out between them (or decide independently in the absence of cooperation) how much each will pay. The general rule (that's legal theory, and it can be displaced by statute or company policy; or by an insured lying on his claim form and just perhaps getting away with the fraud) is that each deals with a proportion (presumptively half) of the loss -- within the terms and limits of the coverage underwritten.
So if my annual out-of-pocket ("franchise") is $300 with one policy and CHF 400 with the other, I won't get any payment from either insurer until that insurer's out-of-pocket amount has been met. Then there is the co-payment amount. If it's 10% on one policy and 20% on the other, I am obviously worse off than if the 10% policy underwriter were paying everything. Plus, I (and my ex-employer if the insurance is a retirement benefit) have been paying double premiums without any perceptible benefit for that double payment.
Some kinds of insurance aren't health insurance at all: "cancer insurance" is more like life cover (and Americans including AIG were big and profitable operator in that sector. http://www.ncbi.nlm.nih.gov/pmc/articles/PMC1304745/ -- it is popular in Japan where there is an extraordinary fear of casncer (I am told), and shame as well so not all eligible policyholders and surviving families ever claim). Some hospital cover, especially that sold in the UK is also independent of any actual cost outlay.
Hope that helps. Read your policy, and also the law behind it. "New-for-old" and "stated value" cover are anomalies and (except for the cancer and hospital niche policies I just mentioned) I can't think of anything comparable in the health insurance field, where as in any casualty and non-life indemnity cover you are not supposed to get back more than you lost and you can't collect in full from two different policies even if you paid full premiums for both. (Distinguishing, of course, where an owner and a lessee of a building each have fire insurance on that building, and each collects in full when it burns down as each as an insurable interest in it.)
For the rest: the exemption I wrote about has limited scope and is not connected with present immigration status but prior employment with an eligible international employer. I would not try to address the issue of retroactive cover except to say that it is open to fraud since obviously if one contracts or discovers a loathesome or chronic or expensive disease one will obviously opt for the best retroactive cover available.