If someone were to lend me a time machine and ask me to go back and figure out exactly what first set me down my road to dedicated descriptivism, I would first ask them if perhaps there wasn’t a better use for this marvelous contraption. But if they persisted, the coordinates I’d start with would be my elementary school days. I suspect it was some time around then that I first asked for permission to do something and was met with one of the archetypal prescriptions.
“Can I go to the bathroom?”, I surely must have asked, and just as surely a teacher must have answered, “I don’t know, can you?”
The irritation that I felt at this correction was so severe that even though I can’t remember when this happened, nor who did it to me, I still can call to mind the way it made me seethe. It was clear to me that the pedant was wrong, but I couldn’t figure out quite how to explain it. So, at the risk of sounding like I’m trying to settle a two-decade-old grudge, let’s look at whether it makes sense to correct this. I say that the answer is no — or at the very least, that one oughtn’t to correct it so snootily.
Let’s examine the “error” that the authority figure is correcting. Can, we are told, addresses the ability to do something, whereas may addresses permission. Mom said I can count to ten means that dear ol’ Mum believes in my ability to count to ten, although she may not want me to do so; Mom said I may count to ten means that Mum is allowing me to do so, although she need not believe that I am able to.*
At any given time, there are a lot of things that one is capable of doing (can do) and a lot of things that one is permitted to do (may do), and a few things that fall into both categories. The prescriptivist idea is that there is a fairly clear distinction between the two categories, though, and so it is important to distinguish them.
Except, well, it’s not so important after all; can and may were tightly intertwined in early English, and were never fully separated. The OED lists an obsolete usage [II.4a] of may as meaning “be able; can”. This is first attested in Old English, and continues through to at least 1645. Furthermore, may meaning “expressing objective possibility” [II.5] is attested from Old English to the present day (although it is noted as being rare now). Examples of these are given in (1) and (2). So we see that may does not always address the issue of permission, that may has encroached upon can‘s territory at times in the past and continues to do so to this day.
(1) No man may separate me from thee. 
(2) Youth clubs may be found in all districts of the city. 
As for can, there’s no historical evidence I found of it referring to permission in the distant past. Back then, may was apparently the dominant one, stealing usages from can. The OED gives a first citation for can meaning “to be allowed to” in 1879, by Alfred, Lord Tennyson, and does call the usage colloquial, at least on the British side of the pond. But still, we’ve got it attested 130 years ago by a former Poet Laureate of the UK. That’s a pretty good lineage for the permission usage.
Furthermore, I think (at least in contemporary American English) that the may I usage is old-fashioned to the point of sounding stilted or even affected outside of highly formal contexts. Just to back up my intuition, here’s the Google Books N-grams chart comparing May I go and Can I go:
You can see there’s a changeover in the mid-1960s, when the usage levels of May I finish plunging and Can I starts rocketing away. As you well know, this sort of fairly sudden change in relative frequency tends to generate a backlash against the newly-prominent form as a sign of linguistic apocalypse, so there’s no real surprise that people would loudly oppose permissive Can I. As always, the loud opposition to it is one of the surest signs that it’s passed a point of no return. By my youth, Can I was ensconced as the question of choice, and nowadays, I doubt many of our kids are getting being corrected on it — though it remains prominent enough in our zeitgeist to function as a set-up for a range of uninspired jokes.
So historically, what can we say of can and may and permission and ability? We’ve seen something of a historical switch. In the distant past, may could indicate either permission or ability, while can was restricted to ability. Over time, may‘s domain has receded, and can‘s has expanded. In modern usage, can has taken on permission senses as well as its existing ability senses. May, on the other hand, has become largely restricted to the permission sense, although there are some “possibility”-type usages that still touch on ability, especially when speaking of the future:
(3) We may see you at Breckenridge then.
The can expansion is a bit recent in historical terms, but that still means it’s been acceptable for over a hundred years — judging by the Tennyson citation — and commonplace for the last fifty or so. The recency explains the lingering resentment at permissive can, but it doesn’t justify it. Permissive can is here to stay, and there’s no reason to oppose it.**
*: Not to telegraph my argument, but even here I find Mom said I can count to sound more like a statement of permission than ability.
**: I have some thoughts on whether it’s really even possible to draw a clear line between permission and ability — in essence addressing the question of whether the smearing together of can and may is an accident or inevitability. I’ll try to put them together at some point & link to them, but given my history of failing to follow through with follow-up posts, I’m not going to leave it as only a possibility, not a promise.