As far as I know, when overpowering you run the risk of: a.) Overheating and damaging the voice coil b.) Causing excursion beyond the physical limits of the driver, again causing damage.
With subs, I worry more about the physical damage, since my naive understanding is that music signal will have a lower average level (though I honestly wonder if speaker manufacturers account for this when they rate power handling for a driver).
So if my amp is capable of 1000W RMS, its probably not producing that continuously on music (gotta be careful on bass tracks), and therefore I feel less likely to reach the thermal limit. Also, big speakers like subs seem to be a little more forgiving of heat than small speakers like tweeters - but that's just a guess.
Enclosure will likely play an important role, as will the exact driver. However, I would think 1000W (good watts) is okay for a 600W or certainly an 800W RMS rated sub. Its probably pushing it for a 300W or 400W RMS rated sub, though. Generally.
