This result comes from the manner in which the orbitals are determined for particular orbitals.
First, the principal quantum number n is determined. This decides to which shell the orbital belongs. n can have any positive integer value starting with 1.
Next, the angular momentum quantum number, l must be specified. l can be any value from zero up to n-1.
An orbital is a p=orbital if it has an angular momentum quantum number, l equal to 1 (which implies that these orbitals first exist for quantum level n=2, and are found for every value of n after that).
Finally, for determining orbitals, the one remaining quantum number to be specified is the magnetic quantum number, m_l. Like each quantum number, there are restrictions on the values m_l can possess. In this case it is -l, -l+1, -l+2,..., 0, 1, 2, ...l-1.
Therefore, putting all this together: if l=1, (so we are referring to a p-orbital, the possible value for m_l are only -1, 0, and +1. These three possible values create the orbitals known as p_x, p_y, and p_z as the only possibilities, regardless on the value of n.