When are there enough Casimirs?
Here we will only discuss the case of finite-dimensional irreducible representations (irreps) of a complex semisimple Lie algebra $L$.
Recall that the set $Z$ of Casimir invariants is the center $Z(U(L))$ of the universal enveloping algebra $U(L)$, cf. e.g. this Phys.SE post.
OP's question is answered without proof on p. 253 in Ref. 1:
Theorem 2. For every semisimple Lie algebra $L$ of rank $r$, there exists a set of $r$ invariant polynomial of generator $t_a$, whose eigenvalues characterize the finite-dimensional irreducible representations.
Ref. 2 (which is one of the most important books on Lie algebras, at least if one is interested in the proofs) does not bother to mention Theorem 2 explicitly. However, it is possible to string together a set of more fundamental results (and their proofs) from Ref. 2 to get the sought-for result. We outline the proof strategy below.
Recall furthermore that there is associated a root system $\Phi$ to the Lie algebra $L$, and let us imagine that we have picked a base $\Delta$ for $\Phi$. The order $|W|$ of the Weyl group $W$ is equal to the possible choices of (unordered) bases and equal to the possible choices of (fundamental) Weyl chambers.
It is proven in chapters 20-21 of Ref. 2. that a finite-dimensional irrep has a unique highest weight vector (unique up to normalization) with some dominant integral weight $\lambda$. We will from now on denote such irrep $V(\lambda)$. (Ref. 2. also defines a notion of a highest weight irrep $V(\lambda)$ when $\lambda$ is integral but not dominant. Such irreps are necessarily infinite-dimensional, so we will ignore those.) It follows that
Two irreps $V(\lambda)$ and $V(\mu)$ are equivalent (i.e. isomorphic) iff their highest weights are equal $\lambda=\mu$.
As a consequence of Harish-Chandra's theorem, the set $Z$ of Casimirs takes the same value on two highest weight irreps $V(\lambda)$ and $V(\mu)$ iff $\lambda+\delta$ and $\mu+\delta$ belong to the same Weyl orbit,
$$ \sigma(\lambda+\delta)~=~\mu+\delta, \qquad \sigma \in W. $$
Here $\delta$ is half the sum of the positive roots. However if both integral weights $\lambda$ and $\mu$ are dominant, then $\lambda+\delta$ and $\mu+\delta$ must both belong to (the interior of) the fundamental Weyl chamber, so that the Weyl reflection $\sigma={\bf 1}$ must be the identity element. In conclusion, we get that
The set $Z$ of Casimirs takes the same value on two finite-dimensional irreps $V(\lambda)$ and $V(\mu)$ iff their highest weights are equal $\lambda=\mu$.
Harish-Chandra's theorem is proven in chapter 23 of Ref. 2. See also this and this related Math.SE posts.
Example: Consider the Lie algebra $L=sl(3,\mathbb{C})$. The Weyl group is $S_3$. The Lie algebra $L$ has two independent Casimir invariants $C_2$ and $C_3$,
$$C_n ~:=~ {\rm str}({\rm ad} t_{a_1}\circ\ldots\circ{\rm ad} t_{a_n}) t^{a_1} \otimes\ldots\otimes t^{a_n}, \qquad n~\in~ \{2,3\}.$$
Consider the 3-dimensional fundamental representation $F$ and the dual/contragredient representation $\bar{F}$ of $L$, which are non-equivalent irreps. They have highest weights $\lambda=(1,0)$ and $\mu=(0,1)$, respectively. In detail, if $t_a$, $a=1, \ldots, 8$ are generators for $L=sl(3,\mathbb{C})$, then (hattip: Peter Kravchuk)
$$\bar{F}(t_a)~=~ -F(t_a)^t,$$
so that the Casimirs $C_2$ (and $C_3$) take the same (opposite) value on $F$ and $\bar{F}$
$$ {\rm tr}_{\bar{F}}\bar{F}(C_n)~=~(-1)^n{\rm tr}_{F}F(C_n), \qquad n~\in~ \{2,3\}. $$
One may prove that the values are non-zero, so that the Casimirs $C_2$ and $C_3$ distinguish between the two non-equivalent irreps $F$ and $\bar{F}$, as they should.
References:
A. O. Barut and R. Raczka, Theory of group representations and applications, 2nd ed., 1980.
J.E. Humphreys, Introduction to Lie Algebras and Representation Theory, (1980).
It can be proven (Racah's theorem) that the number of Casimir operators is the same as the rank of the algebra (number of simultaneosly conmuting generators). This is at least true for semi simple algebras.