as already indicated, medical and life insurance work differently (in north america).
the system is very simple and works well.
for medical insurance, you can not be over insured. there is no way to profit (unless you lied). every medical insurance application form will ask you if you have any other form of medical insurance. they also ask for the company and the policy # of your existing insurance.
your existing policy is your primary, any policy purchased afterwards becomes a secondary. this allows the insurance companies to co-ordinate and make payments. your primary is on the hook first, the secondary covers anything left over (up to the limits of coverage).
this type of co-operation is very common. as an example consider a working couple. both have group dental plans provided by their employer. each insurer will only cover 80% of expenses. so one company (primary) pays 80% and the other (secondary) pays the remaining 20%. no cost to the working couple.