Would insured parents drop their insurance?

An excellent question was asked in Dan Froomkin's online chat yesterday:

I know Bush said his is a rational administration, but how in the world does he think that the expanded SCHIP will induce people to go off private health insurance to use SCHIP -- that is, not insure the parents, but the children? I have had health insurance for me, my spouse, and my children -- and no way would I stop insuring me and my spouse just to get a freebie from the feds.

Yes. Why would people who have good insurance risk bankruptcy from their own catastrophic illness just to get free insurance for their kids? Doesn't make sense, does it?