Actually there are usually two kinds of failure associated with dynamic loudspeakers. Thermal and mechanical. If you feed a 30 cm transducer designed for a sealed enclosure 50 volts AC at 20 Hz, without a load (known as free air) you can bet it will get damaged quickly. You can take the same woofer and load it in an enclosure and feed it over 100 volts at 45 Hz and it would probably be able to reproduce the sound but the voice coil may over heat. If the former gets too hot, it will lose shape. The varnish may start to bubble and the windings will short out.
Feeding a woofer DC is a good way to do this. Making sure your amplifier has sufficient capacity is a good way to prevent this. If your woofers are of a reflex design, you should incorporate a highpass filter with a fairly steep slope of 24 dB/octave close to the tuning point. Reflex speakers offer little or no over excursion protection if driven with frequencies much below their tuning point. Some manufactures use shorted rings at the end of the voice coil windings to prevent woofers from getting out of control and slapping. Others use a progressive suspension combined with a domed back plate.
Biamplified systems in mobile applications experience the highest failures especially in the hands of people that are new to car audio. They often overdrive the LF amplifier and the clipping distortion is never heard because the HF section masks it. Either that or they don't know the sound of an overdriven amp!

The voice coils in the woofers often burn up because of this.
Cheers!