Certainly clipped waveforms from an underpowered amplifier can produce a signal with abnormally high energy content and burn out voice coils. In that particular scenario my personal experience has been that the tweeters were most often affected first.
But every driver and voice coil has it's current and corresponding thermal limit, and when exceeded, can be burned with a powerful amplifier and volume level, even if it's not clipping.
Even if you can't get your nose close to the voice coil at the rear of the driver, putting your nose right up to the dustcap can often determine if that's where the smell is coming from, even through the dustcap, unless it's poly or aluminum cone with sealed poly or aluminum dustcap (provided you have a reasonably sensitive nose and don't have a bad cold).
Steve