It turns out that that is not how double coverage works. Secondary typically only pays <what they would have paid> - <what primary paid>, so if primary paid as much as or more than the secondary would have, the secondary just takes your premiums and does nothing with them.
In addition to the secondary taking your premiums money and giving small or no benefits for it -- in our case, for a couple of weeks over a year -- both insurance companies now in theory perform "Coordination of Benefits" according to formulae established in your policy documents, but in practice it's taken us well over a year to even start to understand their formulae and most of the time they can't even figure out which of them is primary.
So ... for each thing which would normally be 1 bill, 1 EOB, and a payment from you for whatever's left, you end up with 5-20 bills, 2-20 EOBs and for a substantial subset thereof, 1-30 phone calls to 1-3 organisations, taking up many hours of your time trying to figure it out and argue it, and causing indescribable quantities of stress far beyond the removal of your dependent from one of the policies.
In addition to this, if you have a Medical FSA... you rely on EOBs to prove how much you may claim (what you owe, officially). When you get EOBs from the secondary on which none of the numbers except the original billed amount match, it is very hard to show from the combination of primary EOB and secondary EOB what the final patient payment due was. This involves either more back-and-forth between you and your insurance companies or having to avoid claiming the more weird ones and hoping you have enough simple ones to use up your FSA. Building claims for reimbursement from your FSA becomes a months-long, soul-crushing battle even if your FSA administrator company is user-friendly and super-easy to deal with (which I have always found mine to be).
Things I am incalculably grateful for:
- For 2010, we each have only one insurance company!!! YAY!
- My 2009 FSA has been well and truly claimed, drained, justified by more dollars of expenses than can be paid out so there is no question of my having received any funds from it erroneously.
- The only unresolved bill is from December 2008 and so I don't have to worry about FSAs.
 Neonatal Intensive Care Unit.
 It turns out it probably would have been somewhere a bit north of $1000 with only one insurance plan, so not horrendous.
 ... but it took us 16 months to ascertain the degree to which this is true.
 Unless your primary insurance sucks, in which case why not only get the other one (unless primary is free)?
 ... which is a simple formula.
 ... because you keep getting bills while the service provider and two insurance companies screw around. One or both of the insurance companies will usually not have received anything, or will have muddled whether they're primary or secondary, or will have otherwise have royally screwed up their calculation.
 Every time they screw up an EOB, or you think they screwed one up, you end up with re-issued EOBs, often recombining different sets of services into new and confusing groupings of services from different months and different providers into multi-page EOB hell.
 I may be low-balling this.
 Two insurance companies and one provider. Oh, and one visit to somewhere can result in hospital fees and separately billed physician's fees -- potentially blossoming into 5, 10, 15, 20, ... different services and 2, 3, 4, 5 different providers, so I'm not really going in to the full horror here.
 What you paid (a cheque or a credit card statement/receipt or whatever) is not proof of what you should have paid. The final arbiter of that, for tax purposes (and since it is pre-tax dollars, you must be able to prove that you spent them validly on medical costs incurred during the plan year or grace period) is an EOB.
 Others have had problems with the on-line service or have complained about them not processing things or not registering receipt of things. I'm not sure why, but in all the (many) claims I've made, they've always been quick and efficient and when I once had a very small issue it was quickly solved.