There can be many reasons for needing insurance coverage above and beyond what a single policy can handle, but there is never a time when having two policies of nearly identical coverages will be a benefit. A better idea is to have a single health insurance policy and supplemental policies to handle additional costs such as prescriptions, medical equipment, or long term care. Just buying a second policy could even jeopardize the coverage you already have.
When multiple insurance policies might be applied to the same treatment, there is a tendency to deny coverage based on the assumption that the treatment should be covered by the other insurance policy. In this way, you could find that neither policy will pay for the treatment and be forced to pay out of pocket instead. If you must have multiple health insurance policies, make sure that they do not overlap in coverage descriptions.
On the other hand, a supplemental policy that pays for prescriptions or medical supplies that are not covered by your health insurance policy could save you a lot of money. Even if your health plan pays a percentage of the prescription cost, the supplemental coverage would only kick in at the limit of your health insurance, Instead of having a competing claim, you have an assistive policy that prevents your regular coverage from leaving you holding the bill.
The key is to avoid double dipping your health insurance coverage. Where policies overlap, insurance companies have the option of denying coverage. Since most health insurance policies will overlap, it is important to investigate the policies before you purchase a second one. Having one policy for primary care and another one to provide the things your primary policy does not can be a good way to cut costs on family medical care.