From my understanding 'cutting through the mix' relates to perception that a bass that sounds nice and loud (and clear!) on it's own, can completely disappear when you add a band into the room. If I'm right I think this is normally due to problem frequencies at around 400-600Hz. Most band members have a presence in this region (guitars, snare/toms, vocals etc.) and in a room these soundwaves will naturally interact with one another. 99% of the time, physics is cruel and these signals will be 'out of phase' with one another, effectively cancelling them out. If you have this problem scoop your EQ a little (just 3dB may be enough) and you'll often find you can 'hear' yourself again.
The other option is play nice and quietly, with everyones amps arranged sensibly so they don't interfere with one another & most players can monitor themselves quite happily
N.B. this might be complete bollocks if it is, sorry!