In Why can I not go?, not follows can because questions invert subject and auxiliary: I cannot go becomes Can I not go? Contrast I do not know why I cannot get in, where no inversion happens, so that word order never works as a question.
Why can't I go? means the same; the contraction pulls not into can't, which sounds neutral and everyday.
Picture a locked door in an office. You tap your badge, nothing happens, and you turn to a colleague. The plain everyday version is Why can't I get in? Same door, same facts, just the contracted form most people say.
Now imagine you are writing a formal complaint to building management about that same badge. Why can I not access the server room? sounds measured and deliberate, so it fits.
Keep the same door for the third variant. Why cannot I access the server room? is grammatical but rare and reads as old-fashioned, the kind of phrasing you expect in an older novel rather than an email. So the mechanism never changes, only the clothing it wears.
A short analogy for the mechanism: think of the auxiliary as a bus and the negative as its passenger. In why can I not, the bus moves ahead of I and the passenger waits behind on the curb. In why can't I, passenger and bus boarded as one unit.
One limitation: the emphatic feel of the full order is a matter of degree, not a rule, and many speakers hear why can I not as simply formal rather than stronger. If you are unsure which effect you want, the contracted form is the safe default in ordinary conversation.
One limit: the idea that putting not in its full spot adds emphasis comes from research on a different pattern, negative inversion in sentences like can't nobody make him, where the negative moves ahead of everything. That literature treats the emphasis idea as a belief, not a settled fact, and it studies those inversion sentences, not why can I not. So the honest reading is this: why can I not is mainly a formal word-order choice, not a proven stronger or more emphatic way of asking. It stays fully grammatical, but if you want the neutral, everyday question, the contracted why can't I is the safe default, and why cannot I or why can I not simply signal a formal or old-fashioned register.