There are a multitude of ways we can approach this. There are a lot of classifications and labels for things that can be reasonably marked as wrong in language. For instance, an error might happen in speech or it could happen in writing. It might happen more often in informal contexts or in formal ones prone to hypercorrection. Some errors appear everywhere.
Heck, some errors are actually intentional, and aim to reduce the perceived formality of the message. The errors discussed here are assumed to be unintentional.
This answer classifies the types of errors on what area of language study they concern.
Phonological errors
foon speeding
"phono-", according to LDOCE, is a prefix used for whatever relates to sound. Thus, phonological errors are almost always seen in speech, much less in writing. Notably, sometimes they're intentional, and add humorous effect to what's being told.
"foon speeding" is a classic example of spoonerism, where your mind transposes segments of words or phrases.
Morphosyntactic or Grammatical errors
I can't knows how to correct write this sentence.
Morpho-syntax is just a more formal term for "grammar". This is the type of error we're all familiar with: When either the words aren't formed correctly, or their order in the sentence doesn't sound right.
The sentence above is wrong (ungrammatical) because "correct" belongs to the category of adjectives and it can't directly modify a verb like "write". Also after modal verbs like "can", we should always use a non-inflected basic form of the verb, and "knows" is at the very least non-standard.
Semantic errors (errors of logic or meaning)
The little whale is eating the purple sky.
Sometimes, sentences are semantically erroneous. They don't make sense! Semantics deals with meaning in language. An utterance may be right grammatically, but make no sense at all. And that would mean it's nonsensical, not ungrammatical.
The sentence above suffers from a category mistake. No matter how hard the little whale tries, it won't be able to eat the sky.
Word Choice
A very important part of semantics is choosing words. A lot of mistakes learners (such as I) make are word choice errors that either fall into the semantic, or morphosyntactic category.
Prime examples are "affect" vs. "effect", and "lose" vs. "loose".
Pragmatic errors
A: How does the food taste?
B: I love playing tennis!
Some exchanges contain neither grammatically nor semantically wrong sentences, but in the whole, don't make sense. That's where pragmatics comes into play; study of use of language in social contexts.
When you ask your friend, Alex, "Do you have five bucks?", you're not actually wondering if your friend has five dollars. You're indirectly requesting five dollars from him. That's not what the semantics of the sentence can tell you, but pragmatics. That sentence was used when you're buying a sandwich, and it doesn't make sense to suddenly wonder whether Alex has five dollars or not when you're buying something, unless you want to borrow that money.
Punctuation, the grayer area
I don't understand what's wrong with this sentence,
Another type of error people mistake for grammatical errors is an obvious error in punctuation. Punctuation has always been the grayer area. There is no one universal standard for punctuation, and punctuation marks as common as commas and semicolons are major sources of conflicts and discussions on these matters.
However, like all of the subjective topics discussed everyday everywhere, despite so much gray areas, there are some rules almost everyone and every manual of style agrees with. The sentence above should not end in a comma, but either a period, or an ellipsis, which not everyone agrees what the sign for should be, by the way.
The most common punctuation error, universally accepted to be an error, is plenken. That is, adding an inappropriate space before the punctuation mark, which does not exist in English.
Further Reading
Next time you see a sentence on ELL that quite doesn't sound right, don't immediately comment it's ungrammatical! Here is a list of fun and informative resources if you wanted to read further:
[1]: Speech errors, Wikipedia
[2]: Common Word Choice Errors, University of Wisconsin Colleges
[3]: Colorless Green Ideas Sleep Furiously
[4]: That that is is that that is not is not is that it it is
TL;DR It's fine to use more than one in a sentence.
If you're looking to explain it, note that you have two different subjects for the two verbs: "the reason is" and "it is". That's a good hint that you have two separate verb phrases. The two verbs aren't really constraining each other in any way — they could even be in different tenses.
In fact, you could use as many different auxiliaries in one sentence as you like!
I haven't yet told you that the reason why I've studied English all these years has never been because I've been enjoying it, but because my wife has never been committed to learning French.
Clarification
Note that neither of those instances of "to be" is traditionally considered an auxiliary (but see the appendix below for a different opinion).
The usual auxiliary verbs in English are indeed to be and to have. However, those can both also serve as main verbs: they can carry the main meaning of a verb phrase, not tense/aspect information. That's what they're doing here.
How do you know when they're auxiliaries? Because they will be followed by another verb, to make a compound tense. For example, has followed by a past participle makes the present perfect:
He has called his parents once a week since he moved out.
And is followed by a past participle makes the present passive:
Our little church bell is tolled to mark every wedding in town.
You can see that in your sentence, the only other verbs are "study" and "binds". Neither of them is by "is", so they're not relevant for any compound tense. So you have to be as the main verb.
Appendix
As others have noted, some grammar systems call "be" an auxiliary no matter what. The one @P.E.Dant linked to below seems to be an example. But however you slice the cake, at some point you have to draw a distinction between "to be" as a support for another verb vs. as an independent item ("non-core auxiliary", for example?).
It's worth nothing that in any case, "to be" behaves oddly for a verb in English, and so do similar words in many languages. Here it's a copula: it doesn't express an action like verbs are supposed to do, but declares an identity. You will see different grammar systems analyze it in very different ways.
The good news is that no matter which analysis is correct, you're okay to use more than one in the same sentence!
Best Answer
These two questions reflect somewhat different meanings of BE when used of persons.
When we say of someone that he is sociable (or gracious or funny or a complete jerk) we are speaking about their character, who they are.
But when we say of someone that he is being sociable we are speaking about their behavior, how they are for the time being acting.
Similarly, when you ask someone "Why aren't you more sociable?", with the simple-present verb, you are asking about the simple-present version: "What is the cause of your unsociable character?"
But when you ask "Why can't you (or won't you or don't you—they amount to pretty much the same thing) be more social" you aren't really asking a question, you are challenging their behavior: "You really ought to behave more sociably, and I can't understand why you don't."