When one person carries more than one kind of health insurance, we refer to it as "dual health insurance" or "dual insurance." If you've ever been confused about the "primary insurance" and "secondary insurance" boxes on a doctor's forms, this is when they come into play. Typically, dual health insurance is held by spouses who both have health insurance through their jobs, or by children who have health insurance from each of their divorced parents.
On the surface, this sounds like a great idea, but often it's actually pretty confusing because in order to reap the benefits of dual insurance you, the patient, must coordinate benefits with both companies, and one must agree to be secondary to the other after reviewing each other's policy language and other information. With spouses, the primary insurer is usually the one from the initiating spouse's company (so if you and your husband both have insurance through work, yours will be primary and his will be secondary FOR YOU, but that will be reversed FOR HIM). With children, usually the custodial parent's insurance company will take the primary position, but in cases of joint custody, it could be whichever plan has better benefits or has a lower deductible.
When you submit a claim, you'll send the original itemized bill from your doctor to the primary insurer. Then, after they pay their part, you'll send a copy of the itemized medical bill plus a copy of the primary insurance company's explanation of benefits to the secondary insurer, who will determine whether or not additional payments will be made. As well, both companies will report the division of benefits to your doctor and your insurance plan administrator.
It's important to note that having dual insurance is no guarantee of better benefits. Many insurers - Blue Cross, for example - will only pay as a secondary insurer if their claims adjustors feel that the primary insurer underpaid, and you will pay (via payroll deduction) a fee for having secondary insurance coverage. Before you consider adding your spouse's plan as a secondary insurance company for yourself, check with both companies to see if doing so would be at all useful. If not, don't bother - you'll end up wasting money and getting nothing in return.