You mentioned Texas in your question so here is the Texas Department of Insurance's definition of a non-owner policy - it is insurance coverage that offers liability, uninsured motorist, and medical payments to a named insured who does not own a vehicle.
#1 - Non-owner policies are not available in all states. If you want to find out if your state allows for it contact an insurance company to see if it is quoted in your state or your state's insurance regulator.
#2 - The TX Insurance Commissioner's information sheet on the pros and cons of non-owners insurance states that named non-owner coverage is always in excess of the liability coverage bought by the owner of the car being driven. This means if you have an accident, the vehicle owner's liability insurance pays first, up to the limits of his or her policy. The non-owner coverage comes into play only if the cost of injuries and/or property damage goes over the limits of the vehicle owner's liability policy.
This may differ in other States but in general the owner of the vehicle's insurance is primary and the driver's insurance, non-owners or other type of policy, is secondary or excess regarding third party damage or liability.
#3 - If you own a vehicle then you normally cannot obtain a non-owners policy. In fact we are unaware of any insurance carrier that allows you to purchase a non-owners policy in a state that the named insured is a vehicle owner. A non-owners policy can be helpful for a person that does not own a car but frequently rents cars, regularly borrows cars from friends or when driving an employer furnished vehicle.