It is definitely possible to get a second, or even a third health insurance plan. Indeed, it can be a great way of getting additional or more comprehensive coverage. It can also serve as a backup to an employee health insurance, and can reduce the risk for those with pre-existing conditions.
However, it is important to remember that having more than one plan does not mean that you get reimbursed twice for a medical treatment. Even when you have more than one plan, the total amount that they will cover at the time of a claim will not exceed 100% of the treatment cost.
For example, if you get a surgery, you will choose one plan to pay for this up to its coverage limits. Once this insurer has paid the amount, the remaining bill goes to your second or third insurance, which will then cover part of or all of the remaining costs, again up to its coverage limit.