Why is “accidentally” pronounced “accident-ly” instead of “accident-tal-y”?

Solution 1:

For precisely the same reason as UK speakers often drop an unstressed syllable in words like medicine and secretary, making those come out as though they were spelt “medcine” and “secretry” instead.

It is because we sometimes reduce unstressed syllables not just to ambiguity, but to oblivion.

Edit: Barrie notes in a comment that the OED allows for only four syllables in accidentally, never five, so this is hardly uncommon.

Adding laboratory and territory to the list along with medicine and secretary, Patricia T. O’Connor observes here and here that dropping the penultimate syllable is a relatively “new” phenomenon in UK English, arising in the late 18^th and early-to-mid 19^th centuries. (But what happens with words like Worcester is something else, as she shows here).

But it is really no different than people pronouncing centuries as sentries, or -ally words as though they were -ly. Unstressed syllables are simply fragile. It is what happens in contractions, whether written or spoken. It can be also seen when Vulgar Latin developed into modern Romance, so it is not as though this were unique to English.