Your typical EQ uses IIR (infinite impulse response) filters - which are used commonly because of their speed. But they have a non linear phase response, meaning that they alter the phases of different frequencies in the spectrum. And so if you split your signal into n bands using an IIR filter, and then reconstruct it - some of the frequencies will cancel each other out, because of the phase differences, leaving you with a fucked up signal. The proper way to split frequencies is to use a linear phase EQ, which uses FIR (finite impulse response) filters - much slower but they preserve the phases.
First one is a casual square wave, second one is a square wave after being split into two bands at 600 hz and then combined together:

Edit: I should have also said that FIR filters add delay to the signal.