It is often quite possible to have two health insurance plans at the same time. Most often this is the case if you have your own plan and are on your spouses plan as well. However, it is rarely cost effective to do so. While you will be paying both insurance companies premiums based off them paying their portion of your coverage, they will not pay over the total cost of the procedures or visits. Most of the time, you will be overpaying the second insurance for the coverage that they are giving you.
The exception to this would be a plan that is designed to be a secondary insurance plan to cover the deductibles and/or copays of the primary plan.
The exception to this would be a plan that is designed to be a secondary insurance plan to cover the deductibles and/or copays of the primary plan.