The Latin origins of this word (in a verb meaning "happen") dominate its earliest use as a noun in English, where it means simply "occurrence".
There is no suggestion of a disaster, which is the most common modern usage. So when Oberon hopes that the lovers will "think no more of this night's accidents" (A Midsummer Night's Dream, IV. i.67) they are simply talking about events that took place, whether good or ill.
The linguistic context often indicates the positive meaning: Pucelle asks her spirits to "give me signs of future accidents" (Henry VI Part 1, V.iii.4) and Prince Hal says that "nothing pleaseth but rare accidents" (Henry VI Part 1, I.ii.205).
The nearest we get to modern usage (but without any sense of "fortune" or "fate") is when the word is used in the sense of "chance". "By accident, I had a feigned letter of my master's Then in my pocket" says Pisanio (Cymbeline, V.v.278), and Lucius, earlier in the scene (76), tells Cymbeline "the day Was yours by accident".