What would prevent object from experiencing an infinitely powerful jerk in an instant?

^{Note: The following answer has been given from a classical mechanics point of view. It does not deal with any quantum mechanical phenomenon arising due to the following scenarios.}

Theoretical analysis

An infinitely powerful impulse can, theoretically, exist. Because, mathematically

$$J=\Delta(mv)=\Delta p=p_{\rm final}-p_{\rm initial}=p_{\rm final} \qquad \text{(when }p_{\rm initial}\text{ is zero)}\tag{1}$$

where $J$ is the impulse applied. Now since the momentum considered here has to be relativistic momentum, thus the formula $(1)$ gets modified to

$$J=\gamma m v=\frac{mv}{\sqrt{1-\frac{v^2}{c^2}}}\tag{2}$$

Thus, according to formula $(2)$, as $v\to c$, the momentum (and thus the impulse needed) blows up to infinity. Thus you could easily apply an near infinite impulse and still never reach the speed of light. Now since the force applied can be written as

$$F=\frac{ \Delta p}{ \Delta t}=\frac{J}{ \Delta t}\tag{3}$$

So if you manage to accelerate an object to a very high (near light) speed in a finite interval of time, you would be applying as large force as possible. Thus, the closer the object's speed reaches to the speed of light, the larger the impulse (and the corresponding force) you'd need to apply.

However, since an object can never attain a speed equal to $c$, thus the impulse can also never reach infinity. Thus, mathematically $J\in[0,\infty)$.

Practical analysis

Time

^{Note: The following analysis might sound a bit philosophical/metaphysical, however I see no reason for it to be wrong}

Now if you're wondering that if we could make the $\Delta t$ extremely small, to make the force infinite, then... No. Not really. Time, as we perceive, is continuous. You cannot choose any two different unique time instants separated by a zero time difference. It's the same as saying that there are no two distinct real numbers whose difference is zero.

You might argue that why don't we choose the same time instant instead of two different time instant. This is not possible under the domain of classical mechanics, since one of the fundamental assumptions of classical mechanics is that the state of any system is unique at a certain instant of time. Thus you cannot have finite $\Delta p$ for two same time instants. For more insight, see this PhysicsSE question.

Practicality

Practically speaking, I can quite confidently say that there are no man-made machines which can cause any sort of infinite acceleration, and neither are we gonna make any of them. Despite of the unphysical-ness of such infinite acceleration, I reckon we are ever going to have enough energy to accelerate something heavy even to anything near infinity.