Yes. In fact $\bigcup_{n\geq 1} P_n $ is the set of all primes.  Serre  proved that, for each odd prime $p$, there is some very predictable $p$-torsion in $\pi_{k+(p-1)}(S^k)$, for example. 

There is a very nice theorem of McGibbon and Neisendorfer (proving, I think, a conjecture of Serre) that implies and vastly generalizes this.

> **Theorem (McGibbon & Neisendorfer).**  Let $X$ be a simply-connected finite complex such that $\widetilde{H}_*(X; \mathbb{Z}/p) \neq 0$.
> Then $\pi_n X$ contains $p$-torsion for infinitely many valies of $n$.