Generally, people can manage pronouns pretty well. There's one exception, though, and it results in writers doing Luna's dialogue wrong. I speak, of course, of "thee vs thou".

Quick history lesson before I get to the practical gist of this.

A lot of languages have two different second-person pronouns for various degrees of formality. In Spanish, "usted" is used to address strangers and superiors, whereas "tu" is used to talk to friends and equals. At one point, English had something similar: "you" was reserved for addressing superiors, whereas "thou" was used for pretty much everyone else. Eventually, egalitarianism kicked "thou" to the curb, and "you" became our only second person pronoun (not counting "y'all").

The upshot of this is that thou is weird, and not to be trifled with. But the real problems come when we bring "thee" into play.

It's really not complicated, but because of thou's relative rarity, it's frequent that people don't take the time to learn. In 99.9% of things written, that's not a problem. However, when you're writing Luna's dialogue, you have to be careful.

This is because there's a difference between "thou" and "thee". It's basically the same situation as "I" and "me". When the subject of your sentence is the "second person," you use "thou", and when it's not, the correct pronoun to use is "thee".

Here's an example.

Thou hath failed to use these pronouns correctly. For this insolence, they shall destroy thee.

It is also worth mentioning the differences between "thy" and "thine". From what I was able to gather, the main difference is the same as "my" and "mine". In addition, with thy and thine, each was used in a manner similar to that of a/an, where thy was used before words that began with consonant sounds and thine for words that began with vowel sounds, including words that began with the letter "h".

Example:

Thy sentiment is much appreciated, however, thine heinous crime cannot be forgiven.

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I actually have an entire blog post devoted towards such constructions:

The Royal Canterlot Voice and You: A Quick and Dirty Guide to Speaking Far Too Loudly

There are a number of other things that change when you're writing Royal Canterlot Voice/Victorian English/Old Modern English. For instance verb endings change in all conjugations besides third person singular. They also respect the difference between transitive and intransitive verbs when deciding helping verbs, as most romance languages do.

English is weird. That's part of what makes it so awesome.

I'm glad someone took the time to clarify this. However, I need to point out a (slightly ironic) error:

Your example sentence should read "Thou hast failed to use these pronouns correctly."

Let me add a few more helpful tips to this tutorial:

The conjugation of the word "have" in archaic English is weird for us. Here it is:

I have
Thou hast
He/She/It hath
We have
Ye have (Ye is the plural form of "thou".)
They have

Let's take a closer look at archaic conjugation, especially the -eth and -est endings. Here's the conjugation of the verb keep:

I keep
Thou keepest
He/She/It keepeth
We keep
Ye keep
They keep

Notice that the -eth ending only replaces the case in which the verb ends (or endeth!) with an "s", or, in other words, in the third-person singular conjugation.

The -est ending is tacked onto the second-person singular conjugation, and it has no equivalent in modern English (We just say, "You keep,").

NOTE: Never, ever, ever say, "you keepest." This is wrong, even in archaic English.

Incidentally, good examples of proper archaic usage are preserved in some of Shakespeare's works as well as in the King James Version (the original, not the NKJV or NIV) of the Bible, both of which are easily accessible online should anyone desire further examples.

One last thing that's worth pointing out: In the show, Luna herself screws up on some of these rules, sometimes spouting an anachronistic mix of archaic and modern English. I personally disregard this and amend it to proper archaic English in my writing, but if you really want to be accurate to the show and take that as canon, you must study Luna's dialogue in "Luna Eclipsed".

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Luna herself screws up on some of these rules

Chalk it up to bad writing from the canon-staff. It's fairly easy to dismiss as her adjusting to modern Equestrian; she's messing up modern Equestrian and/or mixing them.

2679664 They are usually pretty good about this. It's only near the end of the episode that her major blunder occurs, which might have been deliberate for character development's sake.