Speakers can be blown if the wrong power amplifier is used and damage can be done by either overpowering or underpowering the amp. For example, if the power rating of the speaker is 300 watts capacity you will need a 300- to 600-watt amp.
Some people claim that too little power can blow speakers. This is absolutely WRONG. Too little power will only cause the maximum output level to be low.
Abuse and the defective 'wing nut' (an idiot) connected to the volume control blow speakers with low powered amplifiers. If driving a speaker with low power would cause them to fail, speakers would fail every time you lower the volume on the head unit. I will try to explain what happens when speakers are driven with badly clipped signals (minor clipping is not harmful).
This page deals mainly with speaker damage that involves thermal damage of the voice coil. Speakers can also be damaged mechanically by driving it beyond what the suspension can handle. Mechanical damage is generally caused by driving the speaker with too much power but it can also be done when a speaker is in a ported enclosure and is driven with frequencies below the port tuning frequency.
Most of the damage I've seen has been thermal damage to the voice coil. When a woofer is driven with a high powered amplifier to high levels, there will be a significant amount of current flowing through the voice coil. Since the voice coil has resistance, there is a voltage drop across the speaker's voice coil (which the amplifier appreciates greatly :-).
This means that there may be a great amount of power being dissipated (in the form of heat) in the voice coil. When a speaker is driven with lots of clean power, the cone moves a great deal (in proportion to the output voltage from the amplifier). For speakers with vented pole pieces (or other types of venting), this movement forces a lot of air to flow in the magnetic gap (area where the voice coil rides).
When the woofer moves out of the basket, the chamber that's under the dust cap and around the voice coil expands (increases in volume) which pulls cool air into the magnetic gap. When the woofer moves the other direction, the chamber size is reduced and the hot air is forced out of the vent in the pole piece.
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