Classically it was said that measurement leads to a collapse of the wave function. However, if there wouldn't be any limit on the process on measurement itself, strange things can happen, e.g. a particle that is confined to a small region by a very strong potential could by measuring its momentum with high precision be made to be spread out instantaneously all over space. Likewise, if a particle is detected at one location, so its position is measured with very high precision, at any moment after that it can be detected arbitrarily far away.
In Feynman's lectures on physics, he goes into the explicit measurement process in the double slit experiment, where light is used to detect which slit the electron passed through. As long as the photons can detect the slit the electron went through, the interference pattern is destroyed. When the frequency of the light is reduced to diminish the impact, the interference pattern keeps being disturbed, up to the moment when the wavelength of the photons is so large that they are not able to determine which slit the electron passed through.
Likewise, in this answer by John Rennie to the question how it can be that if you measure the momentum of a particle with very high precision its uncertainty may extend over one light year, he explains why (doing this with light at least) a measurement with that precision would take a year.
As another example, in this answer by DanielSank to the part of my question where it was asked how it could be that a measurement of a particle in a box with infinite walls could be made to tunnel out by making a measurement, it was suggested that there is a fundamental limitation on the precision of the measurement.
In all three examples it looks like the restrictions on the measurement process are exactly conspiring to avoid any undesired behaviour to occur. Is there any fundamental reason for that?