If not, then why? It seems like you could save alot of money and space running active 3 or 4 way off a single amplifier.
Because in the purest sense, "active" is one channel of input and one channel of output. One channel of amplification per driver.
Now, take an active crossover and put it in place of the passive in the diagram. What's the difference? Even if you can adjust the crossover points, it's still technically passive (or "quasi-active"). You'd still be feeding the amp(s) a full range signal, then it passes through the XO/processor where it's split. One channel of amplification per driver is not present.
I'm not saying it's not worthwhile to pursue--in fact, why not use a 2-way + sub active headunit then build a simple passive to highpass the tweeter at like 8kHz? Driver's don't need time alignment over 4kHz. Or, and I've thought of doing this, would be to split the "sub" signal using the amp's XO...so the HU gets set to 200Hz lowpass (or whatever), you split the signal at the amps, sub gets an 80Hz lowpass from its amp, midbass gets an 80-200 bandpass on its amp. Midrange is set to 200-8000Hz in the headunit, tweeter gets set to 8kHz highpass in the same way. Just an example or two...