Susskind & Hrabovsky: For any system $F_i=-\partial_{i}V$? In the following $\left\{x\right\}$ means a configuration point in $3N\text{-dimensional}$ configuration space.  Each $x_i$ represents one coordinate of one particle of the system of $N$ particles.
In The Theoretical Minimum: What You Need to Know to Start Doing Physics
By Leonard Susskind, George Hrabovsky, Lecture 5, p. 100, Equation (5), the reader is told:

It is a basic mathematical expression of one of the most important principles of physics:
For any system there exists a potential $V\left(\left\{x\right\}\right)$ such that
  $$F_{i}=\frac{\partial{V\left(\left\{x\right\}\right)}}{\partial{x_i}}.\tag{5}$$

I don't find that assertion objectionable, but it seems like the type of statement which, had I made it, I would be told that I don't know what I'm talking about.
Quite often we find discussions which separate forces expressible as the gradient of a potential from "non-conservative" forces.  But that may be more a matter of convenience than a valid segregation of forces in Nature.
Is the above quoted statement a generally accepted principle of physics?

I said that I don't find the quoted statement objectionable; but that is not entirely true. It implies that Nature is fundamentally divisible into particle and void.  An assumption that I do not make.  But this is the realm of metaphysics.
 A: It should be noted that Susskind & Hrabovsky a few paragraphs before OP's quote write the following disclaimers:

It is quite possible to imagine force laws that do not come from differentiating a potential energy function, but nature does not make use of such non-conservative forces.

and

It is generally not true that if you have a set of functions $F_i(\{x\})$, that they can all be derived by differentiating a single function $V(\{x\})$. It would be a brand-new principle if we asserted that the components of force can be described as
  (partial) derivatives of a single potential energy function.

It seems that Susskind & Hrabovsky merely try to convey that conservative forces & potential energy are useful principles, especially for fundamental forces and systems where energy is conserved, not that one cannot construct counterexamples.
A: Since force is dimensionally the gradient of energy, the statement is correct if F denotes a force and V denotes an energy. Usually V denotes a potential, in which case F should mean some kind of field. Also, F should be a tensor of one rank less than V. 
Inconclusion, this statement is vacuous.
