That's not likely. Others have already pointed out that the thermal limit isn't what matters, the mechanical limit is. Even if the amp can't push the driver past xlim if it drives it past xmax the distortion will be pretty bad. To handle 1200w without going past xlim a 15 inch driver would need about 14mm xmax, which is very rare with pro sound drivers. Even the 1200 watt 18 sound 15LW2400 has only 10mm.The 15" sub sitting right here by me fires 1200w at will and even in rough conditions never shown discomfort.
The size of the amp doesn't mean you're using all of its power, it just means how much it might be capable of with speakers that won't reach their mechanical limits first.
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) and we'll conclude class with the ever-reoccurring question about whether one can blow speakers up by under-powering them (spoiler alert - yes, you can!)