How our body "works"
Let's figure out what factors does our ability to do something depends on. Let's say you had to lift a $10 \:\rm kg$ weight and a $100 \:\rm kg$ weight upto the same height (and starting from the same height) in about ten seconds). Without getting into the specific numbers, we can conclude that lifting the heavier weight is definitely going to cost us more energy and thus it's expected to be tiresome. Thus, we get our first factor as load.
But now imagine if you only had to lift the $100 \:\rm kg$ block (to the same height), however, you have to do it now by lifting it $1\: \rm cm$ every hour until the block reaches that height$^{\dagger}$. This would definitely be less tiring than the first case. But according to physics, the work done in both the scenarios is exactly the same (ignoring any other activities you did during that boring hour ;)). Which indicates that load isn't sufficient enough to decide the feasibility of doing something. And thus we get another factor, time.
Now both of these factors don't affect our capacity in the same way. Increasing the load makes the task harder (in this instance, to lift the load to a certain height). On the other hand, increasing the time during which the task is to be performed, makes the task easier. So there's some vague sort of relationship such that
$$D\propto F\qquad \text{and} \qquad D\propto \frac 1 t$$
where $D$ is the difficulty of doing the task, $t$ is the time duration and $F$ is the load. But this sort of expression is very similar to the expression of power
$$P=\frac{W}{t}=\frac{Fd}{t}$$
where $P$ is the power, $W$ is the work done and $d$ is the distance across which the force acts. (Do note that the above expression is a crude simplification of the correct power expression). So, we can satisfactorily say that power is quite a nice measure of how infeasible any task is.
Questions
What are the restraining factors? Is it more a matter of frequency of micro-impulses? A failure in coordination? Do the cells fail to keep up with their own speed?
So, now whenever we jump, we usually take a quarter of a second to accelerate ourselves. And let's say we jump up at the speed of $10 \rm m/s$ (it's quite fast, just for context, it's about equal to the average speed with which Usain Bolt ran his record 100 metres race). Also, let's assume the person to person to have a weight of $80 \rm kg$. Computing the power required to do this (assuming no losses):
$$P=\frac{W}{t}=\frac{\left(\frac 1 2 mv^2\right)}{t}\approx 16000 \text W$$
Just to get a feel for how big this power is, the average power consumed by a water boiler is $8000 \:\text W$. So you can power up two such water boilers with the power you require to jump with a speed of $10 \rm m/s$.
Do note that in the above expression, $P\propto v^2$ which implies that the power varies quadratically with the jump velocity. This also implies that an increase in a large velocity by a certain amount will require a higher increase in power than the same certain increase in a small velocity, or in other words, it becomes harder and harder to increase the velocity as the velocity gets larger. Mathematically, this is true because $\displaystyle \frac{\mathrm dP}{\mathrm d v}\propto v$.
Now, you can see the effect of time factor in the following scenario. Imagine climbing up the stairs to reach a height of $100 \:\rm m$ in 5 minutes. Easy enough. Now imagine jumping and reaching the height of $100 \:\rm m$. Impossible! But, the interesting thing is that in both the cases the difference between the initial energy (at the ground, before jumping) and the final energy (at the momentary rest on the top) is the same, which implies, we did the same amount of work, or expended the same amount of energy in both the cases. But in the jumping case, we had to do it within a quarter of a second, whereas in the case of climbing the stairs, we did it over a 5 minute period.
Where does the energy released to produce the force lifting the weight go if it's not converted into kinetic energy as much as it should without weight? Into heat?
Well, the most of the energy expended got used up in increasing the gravitational potential energy of that body. There are also other biological losses, but since this is a physics answer, I might not deal with them here :-) So, ideally speaking, all the work that you did was used to lift the body up and, thus, increase it's gravitational potential energy.
$^{\dagger}$You don't have to hold the block (after lifting it) for an hour. Lift it a cm, put it at a platform at that height, relax. Lift it another cm after an hour, put it at a platform at that height, and then relax. Repeat this. This process is analogous to climbing the stairs, take step, relax, take another step, relax, and on and on. Whereas the instantaneous lifting is similar to jumping.