You are likely overdriving the speaker and the thermistor is cutting out (as it should). It is there to protect the voice coil. Modern amplifiers are way stronger than the 25W, 35W systems many of these speakers were designed to handle. Back in the day, most of the audio people I know wouldn't play their systems above 50% volume because of distortion and risk of damaging speakers - even though speakers were rated at the same rms wattage as the stereo. The problem, in theory, that distorted (non-sine wave) signals can go well beyond rated rms power.
Going back a
GREAT many years I repaired some speakers for a local doctor, Dr. Holden (I can still remember his name, even though it's 40+ years ago).
The speakers were expensive Bowers & Wilkins (B&W), and back then (before they started making their own drivers) these ones used EMI 13x8 bass units. As I recall he had a Leak or Wharfdale amplifier (I didn't have to mend that) rated at a conservative 35W, and the speakers were only rated at 15W or so.
Unfortunately he went on holiday, and his teenage son had a party - I think you can see where this is going
So I had them for repair, both bass units killed, and I contacted Wilslow Audio (who were, and still are, a UK speaker specialist) to be told the 13x8's were no longer available - and what you had to do was cut the cabinets out and replace them with EMI 14x9's, which also had the benefit of higher power handling. So I cut the cabinets, fitted the drivers, sounded great again.
So exactly like you said, it worked perfectly fine - while been used 'properly' by the owner, but not when been abused by his son (and friends) at a party.
My issue with the OP's problem is how loud is he driving these speakers? - and is he really overdriving the amp enough to generate so much distortion that the 'fuses' trip out?. It's difficult with out seeing and hearing the problem, but i would associate pops and crackles more with the bass units than the tweeters?.