It depends on how they are designed but... basically... yes.
It's easier for a network filter to deliver different frequencies to different drivers. There are designs that allow signal frequencies above a certain point to flow more easily to one driver and signal frequencies below a certain point to more easily flow to another driver.
In this case, the crossover filter network will not have to dissipate as much signal as it would if there is an independent crossover filter for each driver.
In the case of putting a capacitor on a tweeter, bass energy the tweeter isn't receiving is being absorbed in the capacitor and radiated as heat. If you were to remove the capacitor, you would find your tweeter will get really hot. The power has to go somewhere.
The capacity of the components is not the only factor in determining a crossover network's ability to handle power. The value of the components has a lot to do with it too.
In a 6dB Butterworth crossover design, it's pretty straight forward. You have a crossover point and all frequencies above or below (depending on the configuration of low or high pass) are basically shorted out. That is to say, signal at those frequencies is provided an easier return path than through the speaker. In this case, the crossover has to dissipate all of that energy. Still, the crossover point will largely determine how much power the filter network has to dissipate.
With a 12dB crossover (2 step ladder), the amount of energy each component has to dissipate is different again. 18dB would be your 3 step and 24 dB would be your 4 step ladder. ..... at least in a Butterworth design.
In other words, the number of turns on the inductors, as well as the value of the capacitors has a huge impact on the amount of power the filter is required to dissipate.... the power rating of the components has an impact on the amount of power the filter can dissipate.
With an active crossover, the same principles apply but it's all done on such low power signal that energy dissipation is not much of a problem.
The benefit of an active crossover is that downstream of the active filter, the amplifier doesn't have to amplify a bunch of signal that is only going to be absorbed by the crossover filter. That means less work and energy required of the amp... and no need to short out a bunch of unused signal at line levels in front of the speaker. It's a far, far more efficient way to go. ... but brings with it other problems that can be relevant in some situations.
Nice post Tom. Lots of good info.
Here's another thought to ponder. On my Focal Utopias, I'm running bridged 4 channel Fosgate Power 800's. So bridged they are about 500 watts per channel at 4 ohms. That allows me to have enough power for each set of mids to keep the gains at a minimun level and keep the distortion to a minimum.
The bad part is that when I first built the system I was destroying my tweeter crossovers and blowing tweeters left and right. I'm still running the Focal tn 51 tweeters but I had to go to a third order crossover with some resistors because the tweeter couldn't handle the power. I also ran two sets so that the tweeters are 8 ohm, to help handle the load. So one amp has two sets on it and the other has no tweeters on it. This made a huge difference in reliability. As for building my crossovers I have some audiophile nerds that deal in home audio build them for me. I have no idea how they come up with the stuff they come up with.
On Infinity crossovers, I have a set of Kappa's in my truck, that were in a customers boat but kept throwing his amp into protection mode. I put them in my truck with a JL amp and wouldn't you know it, same thing. I switched the crossovers out for some old Focal crossovers I had stashed for a rainy day and no more problems.
Those Infinity's gave me fits, in the boat and in my truck. I guess the amps were seeing a short when they would get too hot or the volume level got too loud. What a pos they are (the crossovers). The speakers themselves actually sound pretty decent and look well made though.