Don’t confuse speaker breakup with “breaking in” a new speaker — they’re completely different things! Often when you buy a new speaker that hasn’t had audio pushed through it yet, the cone will be stiff, meaning that it won’t vibrate quite as freely as a “broken in” speaker. Many guitarists like to “soften up” or “break in” the cone by playing audio through the speaker; anything from an album playing on repeat to white noise can be used, although simply playing through the speaker is arguably the best. The purpose of this is to get the cone accustomed to vibrating back and forth, which some claim can make it more responsive and sweeter sounding.
“Speaker breakup” refers overloading the speaker so that it distorts. In some cases, this can sound “crackly” and muddy rather than tuneful. However, certain classic tones have been created by overloading the living daylights out of speakers. Use this technique at your own risk, however, because too much speaker overloading will definitely shorten its life or potentially even blow the speaker.