It is often the case that the VC-dimension of a hypothesis class equals (or can be bounded above by) the number of parameters one needs to set in order to define each hypothesis in the class.
For instance, if $H$ is the class of axis aligned rectangles in $R^d$, then $VCdim(H) = 2d$ , which is equal to the number of parameters used to define a rectangle in $R^d$.
How do you understand such a phenomenon?
The intuition that VC-dimension is simply an instance of "parameter counting" is wrong.
Below is a famous counter-example. For any $t \in \mathbb{R}$ define a function $f_{t}$ taking values in $\{-1, +1\}$ as follows: \begin{align*} f_{t}(x) &= \begin{cases} +1 & \text{if }\sin(tx) \geq 0,\\ -1 & \text{if }\sin(tx) < 0. \end{cases} \end{align*}
Consider a concept (or hypothesis) class $$ \mathcal{C} = \{ f_{t} : t \in \mathbb{R} \}. $$
Then for any $d \in \mathbb{N}$ and any sample $x_{1}, \dots, x_{d} \in [-1, 1]$ the concept class $\mathcal{C}$ can achieve any labelling of the given sample. The key idea is that for large values of $t$ the function $\sin(tx)$ oscillates at a very high frequency. Since we are free to choose this frequency, any labeling of the sample $x_{1}, \dots, x_{d}$ is possible to achieve.
Hence, we have constructed a concept class of infinite VC-dimension which is parameterized with only one parameter.
