This is a cliché of uncertain origin, and there is truly nothing to support that a third time should be any better than the first or the second, or that a fourth attempt may not be necessary.

Contrary to popular belief, the phrase does not date back to English common law that states if attempts to execute somebody fail three times, then that person is pardoned, or at least not executed.  Even though there was a recorded case in 1885 of a person whose death sentence was commuted after three attempts failed, that was just an isolated decision, and not one based on any laws, nor were any laws enacted because of that.  (Martin)

The first recorded instance of the phrase seems to come from the Bard himself.  In the first lines of Act I, Scene Five of The Merry Wives of Windsor, first published in 1602, Shakespeare has his character Falstaff state, “Prithee no more prattling.  Go, I’ll hold.  This is the third time; I hope good luck lies in odd numbers.” (Origin of the phrase)

The specific phrase “third time’s a charm” first appears in The Cabinet Album, which was a collection of various stories and essays published in London in 1830.  And the phrase “third time’s lucky” first appeared in 1833, in William Johnstoun N. Neale’s The Port Admiral.  (Origin of the phrase)  Now there’s a volume I’m sure we’ve all read.

The phrase, more or less, can also be found in Elizabeth Barrett Browning’s Letters Addressed to R. H. Horne, which was published in 1839.  There she states, “‘The luck of the third adventure’ is proverbial.”  And again in 1862, when Alexander Hislop comments in The Proverbs of Scotland that "The third time's lucky.”  (Martin)

The first time the phrase appears in American Literature, specifically as “the third time is a charm,” is in June of 1912, when The Weekly Sentinel reported on a woman’s being granted a license for her third marriage.  (Martin)

Aside from early literary references, the number three has often been seen as a lucky number, unless, of course, you’re talking about dying, when it is said that “Death always comes in threes.”  (Shelton)  It is unclear, though, why Death prefers the number three.  Three has specific meaning in Christian mythology, such as with the Trinity.  And when doing comedy, three has long been considered the limit for a joke.  If the audience laughs at it once, do it again.  If they laugh a second time, go ahead and try it a third.  But after that, quit while you’re ahead.  And the next time you’re watching somebody fumble with their keys in a movie while the killer approaches, count how many times they get the wrong one.  (Martin)

Of course, this is all supposing that luck has any power to change the events in our life to begin with.  It is supposing that, regardless of your effort, you will magically be granted success just because you’ve tried the prerequisite amount of times.  And that is just plain silly.