I test flew a model (his own design with a really big really far forward horizontal tail to get it in the strong downwash) for a friend and found this happened during the flight. It was impossible to keep the airplane level and the flight looked like a smooth up and down motion (or a sine wave for the more math inclined) but large in amplitude. I was dreading trying to land it because he was at the time a really good friend and I hated to crash the monster on the first flight. Gross.

When I throttled back the airplane stopped the motion and started to fly like a stable airplane, touchy but stable. It turned out that it was just a hair on the unstable side when I took off with full power and removal of the power and it's destabilizing effect made the airplane OK.

See if the Cub has the same effect in glide as power on. If no change then the CG is probably OK, If the effect goes away in glide then the CG is just too far back. Also just be sure it isn't the good thumbs of the pilot that is making the plane OK in normal flight.

If the CG is OK then look for a mechanical issue, pushrods, servos, etrc as noted by the excellent writers above and or flexing of the horizontal tail with elevator deflection at high speeds. That will cause a twist that will give a load opposite to the elevator deflection. I have some foamys that act this way.

An also had foamys that the pushrod flex cause the same effect at high speeds. It is one of those things that you can find out by just loading up the elevator by hand and seeing what happens to the normally straight line that the pushrod should be.