Can the IPA, or any other formal linguistic system, fully characterize every accent discernable by language speakers? By "accent" I mean regional variations in pronunciation – for example, the nuances that can distinguish an English speaker from Boston vs. New York vs. Philadelphia.
It cannot fully characterize all differences discernable by speakers (I'm being more general that just dialect differences). The system does, however, (purportedly) have a system for representing at least the phonemic contrast of any language. For example your language has the vowel phonemes /ɪ ɛ ʊ/, as represented in the IPA. The actual way that you, specifically, pronounce those vowels at a particular time requires a finer granularity than exists in the IPA. There are a set of "nudging" diacritics that allow you to say "a bit higher" or "a bit lower" (compared to the standard), but this still does not provide the necessary degree of granularity needed to describe the myriad dialect pronunciations of /ɪ/ in English alone.
A basic reason for us lacking this ability is because of the under-studied notion of "discernability". Linguistics generally thinks of speech perception in terms of "perception within a language", for example whether you can hear the phonemic contrasts of a given language. The test is to present stimuli that are ambiguous and ask speakers to classify words, by asking whether the stimulus was "pit" versus "pet". The choices have to both exist within that linguistic system. In principle, one could devise tests that ask speakers to classify a vowel as being "like the German vowel" or "like the Hindi vowel", but we don't do that. Instead, we measure the objective acoustic properties of phonemes in different dialects or languages, and bypass the step of perceptual / categorial analysis (where you ask "is this the same vowel, or a different vowel"). Since experimental subjects will invent all sorts of understandings of what it means to be "the same vowel" vs "different", we don't ask those sort of questions (is "ea" in leak the same vowel as "ee" in leek? Yes or no, depending on what the subject thinks you mean by "vowel" i.e. pronunciation vs. spelling).