Ever since I started practicing philosophy as the art of life, and teaching people about it, I have operated under two fundamental assumptions: (i) such art can be learned (or I'd be wasting my time trying); and (ii) it can be taught (or I'd be wasting my time writing and speaking about it). Since it is always good -- from time to time -- to pause and reevaluate one's assumptions, let's see how these two hold up to scrutiny.
Socrates famously held that wisdom is a type of techne, that is, a skill, analogous to shoemaking, or crafting musical instruments, or the art of war. It follows that it can both be taught and learned, though he was famously skeptical of those who alleged to be able to teach it, like the Sophists.
Before we look more carefully at the analogies that Socrates drew, it is worth trying to ask what exactly wisdom is. Dictionary.com provides a useful starting point:
"Wisdom, n.: Knowledge of what is true or right coupled with just judgment as to action; sagacity, discernment, or insight."
So wisdom is a particular kind of knowledge, just as Socrates maintained. But it isn't just factual knowledge, although some degree of factual knowledge about the world is necessary in order to live wisely. It is, more properly, a moral understanding of things. A wise person is able to navigate the world better than an unwise one not just because she knows things, but because she has reflected on the value of those things and has decided to act accordingly.
A temporal component also seems a necessary -- but, crucially, not sufficient -- ingredient for wisdom. Children are not wise for the simple reason that, no matter how smart they may be, they just don't have enough life experience to be able to assess things properly. Then again, just getting older doesn't ipso facto give you wisdom: you have to be able to critically reflect on and learn from your experiences, not simply accumulating them passively.
Wisdom, in the ancient Greco-Roman literature, is also divided into two types: theoretical and practical. Aristotle discussed the distinction in book VI of his Nicomachean Ethics. Theoretical wisdom (sophia in Greek) is "scientific knowledge, combined with intuitive reason, of the things that are highest by nature" (Nicomachean Ethics, VI, 1141b). As for practical wisdom (phronêsis in Greek), Aristotle says: "Now it is thought to be the mark of a man of practical wisdom to be able to deliberate well about what is good and expedient for himself, not in some particular respect, e.g. about what sorts of thing conduce to health or to strength, but about what sorts of thing conduce to the good life in general" (Nicomachean Ethics, VI, 1140a–1140b).
The distinction between theoretical and practical wisdom led Aristotle to claim: "This is why we say Anaxagoras, Thales, and men like them [i.e., the Presocratic philosophers] have philosophic but not practical wisdom, when we see them ignorant of what is to their own advantage, and why we say that they know things that are remarkable, admirable, difficult, and divine, but useless; viz. because it is not human goods they seek." (1141a).
I wouldn't go as far as saying that the kinds of things the Presocratics were after are "useless" -- after all, they were the original scientists, interested in foundational knowledge about nature -- but they are not particularly useful when it comes to living a good life. Just because an astronomer knows all about galaxies it doesn't mean that his life is going to be moral, or happy (which, for the Greco-Romans, amounts to the same thing).
The Stoics famously claimed that we need both kinds of knowledge: enough sophia to understand and navigate the physical world, informing our phronêsis. Indeed, they reduced phronêsis to a remarkably simple concept: the knowledge that the only thing that is truly good for us is a good character (and whatever improves it), while the only truly bad thing for us is a vicious character (and whatever makes it worse). The reason being that a good, virtuous character will allow you to use everything else (health, wealth, education, fame and so forth) for good, while a vicious character will do the opposite. Hence, virtue is the only intrinsic good (and vice the only intrinsic bad).
But let's get back to Socrates and focus on one of his analogies to make the point that wisdom is a type of techne: the crafter of musical instruments. People making instruments are not the same as musicians, though of course there may be some overlap between the two. It is perfectly conceivable that a person be an extraordinary artisan, capable of making exquisite musical instruments, and yet be a mediocre musician. It's also the case that most musicians wouldn't know the first thing about making their own instruments. That's because the two sets of skills are broadly distinct.
Similarly with wisdom, especially when understood as the knowledge of how to live a good, moral life. Consider Seneca, for instance. He is famously controversial even within Stoic circles because he talked the talk but arguably didn't exactly walked the walk. Although the situation is a bit more complex, he himself admits as much to his friend Lucilius:
"There is no reason why you should desire to come to me for the sake of making progress. You are mistaken if you think that you will get any assistance from this quarter; it is not a physician that dwells here, but a sick man." (Letter LXVIII.9)
Let's then think of Seneca as highly proficient at teaching, but not so good at practice. Now, for contrast, consider my grandfather Tino. He was a very ethical and wise man, and yet he knew nothing of philosophy, and had never heard of the art of living. So, Tino was very good at practice, but had close to zero theory (except for whatever he picked up from his family and social environment).
I have personally learned a lot from both Tino and Seneca, although in very different fashions. Tino taught me by example how to play the music -- to continue with our analogy -- while Seneca has given me an invaluable conceptual framework to use in order to make sense of my practice and improve it.
I would argue that the best of both worlds is someone like Socrates himself: excellent at teaching, superb at the practice. That is why Socrates is far better known than both Seneca and my grandfather Tino. And that's why Epictetus said:
"This is the way Socrates became what he was, by paying attention to nothing but his reason in everything that he encountered. And even if you are not yet a Socrates, still you ought to live as one who wishes to be a Socrates." (Enchiridion 51)