Two drawbacks. The first is as you suggest: Bags are designed so that the insulation can loft to its maximum, creating "dead air" around your body, which is what keeps you warm. By stuffing one bag inside another, you may compress the insulation of the inner bag to the point where it doesn't do you too much good. Lay one over the other, and use as the inside bag the one that has the slimmest profile.
The other issue is weight. The Andromeda Strain weighs about three pounds, the Cat's Meow about the same. And both are somewhat bulky, although not bad. That will result in six pounds of fairly bulky sleeping bag, which may be more than you want to deal with. By comparison, if you're willing to spend a little money, you could get an L.L. Bean Summit 775 0-degree bag for $285. It weighs just over three pounds, and compresses very nicely. Add a lightweight vapor barrier liner to it (usually about $20) for a bag that would go nicely down to -10 degrees.
Still, for occasional use, I think combining your two bags will work fine.