Thanks to Freudenthal we know that $\pi_{n+k}(S^n)$ is independent of $n$ as soon as $n \ge k+2$. However, I was looking at the table on Wikipedia of some of the homotopy groups of spheres and noticed that $\pi_2^S$ (the second stable stem) and $\pi_6^S$ are achieved earlier than required by the suspension theorem. So my question is:

What is known about this phenomenon of early stabilization in the homotopy groups of spheres? Does it occur finitely many times or infinitely many times? Also, is there any sort of intuitive reason why stabilization might occur early?

Also, I spotted a few times where stabilization almost appeared, but there was a pesky copy of $\mathbb{Z}$ that appeared and then disappeared right before the stable range. I'm talking about the cases of $\pi_{11+n}(S^n)$, $\pi_{15+n}(S^n)$, and $\pi_{19+n}(S^n)$. I assume this has something to do with Hopf elements showing up somewhere?

I know pretty much nothing about this except the basic homotopy theory, a few very small calculations, and the Pontrjagin construction relating all of this to framed manifolds- so any references or illuminating insights would be helpful!