Good question! Short answer: No. It is true for search problems with only one solution but does not hold for search problems with more than one solution. For the long answer, I will need to define some terms (I'll try to use most of the same notation as in the Qiskit tutorial).
Suppose we wish to search through $N$ elements, and that the search problem has exactly $M$ solutions, with $1 \leq M \leq N$. As you know, the Grover iteration can be regarded as a rotation in the two-dimensional space $\mathbb{C}^N$ spanned by the starting vector $|s\rangle$ and the state consisting of a uniform superposition of solutions to the search problem, $|\omega\rangle$. Following Qiskit's naming convention, we will also introduce the state $|s'\rangle$, which is in the span of these two vectors, and perpendicular to $|\omega\rangle$. We can define these normalized states as,
$$|s'\rangle \equiv \frac{1}{\sqrt{N-M}}\sum_{x\neq\omega}|x\rangle$$
$$|w\rangle \equiv \frac{1}{\sqrt{M}} \sum_{x=\omega}|x\rangle,$$
and can therefore express the initial state $|s\rangle$ as,
$$|s\rangle = \sqrt{\frac{N-M}{N}}|s'\rangle+\sqrt{\frac{M}{N}}|\omega\rangle.$$
Let $\sin{\theta} = \sqrt{M/N}$, so that $|s\rangle = \cos{\theta}|s'\rangle + \sin{\theta}|\omega\rangle$. The action of a single Grover iteration, $G$, is to rotate the state vector by $2\theta$ towards $|\omega\rangle$. Based on the initial state, $|s\rangle$, taking the system to $|\omega\rangle$ requires rotating through $\arccos{\sqrt{M/N}}$ radians.
For Qiskit's two-qubit example, we set $N=4$ and $M=1$. This tells us that $\sin{\theta} = 1/2$, so $\theta = \pi/6$, and,
$$|s\rangle = \cos{(\pi/6)}|s'\rangle + \sin{(\pi/6)}|\omega\rangle.$$
From above, we know a single Grover iteration will rotate the state vector by $2\theta = \pi/3$:
$$G|s\rangle = \cos{(\pi/2)}|s'\rangle + \sin{(\pi/2)}|\omega\rangle = |\omega\rangle.$$
So, in this special case, only exactly one iteration is required to perfectly obtain $\omega$. However, this is not the case for all values $M/N$. For example, when $M/N << 1$ we have $\theta \approx \sin{\theta} \approx \sqrt{M/N}$, and thus the angular error of the final state can be as high as (but no greater than) $\theta \approx \sqrt{M/N}$, giving a probability of error as high as (but no greater than) $M/N$. Depending on the value $M/N$ and whether $M$ is known in advance, alternate approaches to and various adaptations of Grover's algorithm are also used. When $M/N \geq 1/2$, we see that the angle $\theta$ gets smaller as $M$ varies from $N/2$ to $N$. As a result, the number of iterations needed by the search algorithm increases with $M$. When $M/N$ is this large, error probability can be as high as (by no greater than) one-half. Here, quantum search offers little significant advantage over a classical one. When $M$ is not known in advance, there are a number of different approaches. One clever procedure exploits the fact that the Grover iteration is periodic, and combines Grover iterations with an application of the quantum Fourier transform to learn the value of $M$ with enough accuracy to still ensure a high probability of success. Nielson and Chuang Section 6.1 and Mermim Chapter 4 are great for more on this topic.
Hope that helps!
