A good way to look at it is considering what the effective impedence will be. Driving a full range signal into, say, a properly crossed over tweeter will result in 'normal' current flow only in the treble range, so taken as a whole (voltage v. current) the apparent impedence of that load would be very high. So your treble amp might see an equivelent (depending on signal) of 20+ ohm load while the mid-bass amp might see a 11 ohm load (just pulling numbers out of air for illustration.)

The amp may run cooler, suffer from less rail sag, etc, but you won't actually get rated output except in limited conditions, such as an input signal that falls completely within it's frequency range. In this case the effective impedence would be back to the expected value.