How does a speaker blow out?

From my experience too low power (ie HF clipping) is generally the culprit behind tweeters failing, but over power (possibly LF clipping but certainly overheating the   voice coil ) is why you blow woofers.  Subsonics can often be the mysterious cause of over output in LF.  Curious  this happened on notoriously difficult  organ material.    

What Erik said (regarding clipping), but: after a x-over passes that high freq distortion to the tweeter and burns it (typically: open), the energy that no longer has a path that way, is routed to the next highest freq driver, in the circuit.

True, but since most crossovers are in parallel, the HF voltage seen by the woofer remains the same. More likely to burn the first coil in the low-pass filter than the woofer itself, but things happen. :)

The amp having less current draw due to an open tweeter circuit could mean a higher voltage though. 

In my case I blew my Tekton impact monitor tweeter because I got so involved with the classical music Iam playing , I turn the volume too much more than the speakers can handle, first distortion then I smell like something is burning. Iam glad Van L Speakers here in Chicago able to fix them.

       After having repaired a few hundred speaker systems; I can only speak from my own experience/observations.

       A clipped signal and it's energy, after opening up a tweeter and having no where else to go, will end up in the x-over's, next-highest freq, driver.

 More likely to burn the first coil in the low-pass filter than the woofer itself, but things happen. :)

       Lots of blown up electrolytics, BUT- in over 15 years of repairing speaker systems (mostly: college student or Pro musician owned, it always seemed): I never once saw a burnt inductor.    Some: with signs of having been abused and accompanied (in series) by a burnt driver voice coil, but- never one that opened or shorted.   

       Perhaps the customers that clipped their amps, were just lucky in that regard.

       Then too: there are a multitude of systems out there, with no inductors in series w/their woofers, to block such damaging, high freq, energy.

                                              YEP (things happen)!

                                                   Happy listening!

As others have said, clipping generally causes damage to the tweeters before the woofers. If you heard a bad noise from the woofer, then it seems likely it is a mechanical failure, not an electrical failure. 

Does the speaker make any sound when you play it now? Is the tweeter and midrange still working? If so, and the woofer is not totally dead, then this seems like it is damage due to overdriving, not clipping. 

You can often tell if there is mechanical damage by gently pushing on the cone. If you feel any scraping or stiction, then you have mechanical damage. 

How old are your speakers? Have you always used the relatively low power tube amp to drive them? Do you use any equalization or dsp to boost the bass?