The following question was put on hold: Is it possible to prove that an elephant and a human could jump to the same height?
It reminded me of an exercise (24a) on that exact topic in Arnold's "Mathematical Methods of Classical Mechanics". The solution he gives goes like that:
For a jump of height h one needs energy proportional to $L^3h$, and the work accomplished by muscular strength $F$ is proportional to $FL$. The force $F$ is proportional to $L^2$ (since the strength of the bones is proportional to their section). Therefore, $L^3h$~$L^2L$, i.e. the height of a jump does not depend on the size of the animal. In fact, a jerboa and a kangaroo can jump to approximately the same height.
The comments of the above question tended to dismiss that argument. What's wrong with it?
Addendum: It seems obvious that not all animals jump exactly to the same height, given their different physiologies/shapes. Some of them can't jump at all.
The question is to be understood in the following spirit: if we plotted jumping height vs animal size for a lot of different species, would there be a correlation? I don't mean there is no spread; I totally expect a big spread due to the other factors involved.
Second addendum: Some interesting points have been raised in comments and answers. I will accept an answer that incorporates: the domain of validity of Arnold's argument (or explain why it is never valid), the effect of air drag on very small jumpers and the impossibility of very large animals.
Bonus points for documenting the yet elusive jumping elephant and plotting jumping height vs size for different species ;)