When I was in second year phonetics in university, our exam was to do just that. Our lecturer would say some sounds, we had to write them down in IPA.
With some provisos.
- Most diacritics would count, but some of them, such as the forward/backward, raised/lowered diacritics for vowels, would not: too subtle. For that matter, we would not be expected to tell apart the 5 different versions of schwa, and I’m not sure anyone does.
- I think we were off the hook for learning the most obscure articulations: epiglottals, alveolo-palatals, and that weird Swedish combination ɧ.
- The consonants were pronounced between vowels: awa, aɥa, aɰa, aca, aka, aqa. That’s optimal for telling the difference between consonants; the auditory cues for the differences are in the transition into and out of consonants. Final unreleased consonants, such as you routinely get in Cantonese, I have found utterly impossible to hear the difference between.
I don’t have a great ear. But under ideal test conditions, and limiting ourselves to distinct IPA letters? Yes. We do.