I think you're missing the context.About twice a year I have a headache, which a couple of aspirin always alleviates. But one time I had a headache and took a couple of aspirin, but an hour later I still had a headache. So the treatment failed.
I took a couple more aspirin. But then I noticed it didn't do anything for my headache. So, the treatment failed.
So I took 3 more aspirin--7 extra strength aspirin within about 2 1/2 hours. Again, the treatment failed. It was intractable pain. Right?
I began thinking that something was terribly wrong and I might have to go to the hospital, that I might have a brain tumor or this was the beginning of a stroke, because it had never happened that aspirin didn't cure my headache.
Then I looked more closely at the bottle, and saw that it wasn't aspirin. I had taken a bunch of melatonin.
But the treatment had failed hours earlier--it was an intractable headache, according to your criterion.
If one has incurable unbearable pain, & considers suicide as preferable, this is reasonable.
An occasional bad headache wouldn't rise to that level IMO.
May you find relief for your headaches.