The need for a 'particle description' of electrons Is there any phenomenon where the 'wave description' of the electron's motion is not applicable?
The reason for this question is to find out if there are any situations were quantum wave theories fail. If not, it seems that the Everett's multiworld analysis as described by Coleman, http://media.physics.harvard.edu/video/index.php?id=SidneyColeman_QMIYF.flv works.
 A: 
Is there any phenomena where the wave description of electron (motion) is not applicable?

Let us touch base and have a look at this bubble chamber picture:

This is one single elementary particle interaction, rare, because it is the creation of an omega particle, but it is one single instant.
There is nothing wavelike in this datum. The particles behave in the magnetic field exactly like charged classical particles are expected to behave and their momentum and mass can be measured or fitted to great accuracy.
This is a particle interaction. The point I want to make is that even though this is a reaction that can only be mathematically modeled statistically by quantum mechanics, there is absolutely no manifestation of the wave nature of elementary particles in this one instant.
We might have a single electron turning in the field . It is not interesting enough to be recorded for teaching purposes, but again it would not display any wave properties.
In conclusion, in the quantum mechanical framework, single events/instances can be described by classical trajectories and physics. It is when the statistics are accumulated that the wave behavior appears. The statistical distribution of such scatterings will be a probability distribution given by the quantum mechanical wave equations, and will display the wave nature of the underlying framework. The waves in quantum mechanics are probability waves . Many instances must be accumulated in a distribution to manifest the wave nature . In the double slit experiment with single electrons  a single electron does not express any wave nature. One can calculate its trajectory classically after the fact. One cannot predict the trajectory unless the probability wave nature of the underlying framework is taken into account.
To summarize for individual measurements the wave nature may not appear at all or cannot be predictive of a trajectory. What the wave nature does is predict a statistical distribution for the particles under consideration. In the example above, a statistical distribution will give the crossection for producing omega- in kaon interactions, angular distributions etc , and these distributions can only be explained with the wave nature of quantum mechanical entities.

The reason for this question is really to find out if there are any situations were quantum wave theories fail. 

The quantum wave theory is continually validated by all experiments. If there were a failure it would have meant a major disturbance in the physics community. Everything is consistent with the particle manifestation for individual events, probability wave distributions for statistical accumulations. 

If not it seems that the Everett's multiworld analysis as described by Coleman, http://media.physics.harvard.edu/video/index.php?id=SidneyColeman_QMIYF.flv works.

This last is not clear, seems to me a non sequitur .
A: The physics of particle mechanics is contained within quantum mechanics, so I don't know if it's correct to say that there is a place where the "wave description is not applicable." However, there are experiments where the wave description is not apparent.
The beginning paragraphs of this page give a very short historical account of wave & particle behaviors of particles. Here's a quick example:

Certainly the early experiments on the properties of electrons did not suggest that any unusual behaviour was to be expected. Everything pointed to the electron being a particle of very small mass. The trajectory of the electron can be followed in a device such as a Wilson cloud chamber. Similarly, a beam of electrons generated by passing a current between two electrodes in a glass tube from which the air has been partially evacuated will cast the shadow of an obstacle placed in the path of the beam. Finally, the particle nature of the electron was further evidenced by the determination of its mass and charge.

Those three examples suggest particle-like behavior. But keep in mind that quantum mechanics, and its inherent wave-like nature, is capable of accounting for the observations.
I haven't read the entire page I linked to, but some of the wording I saw is very carefully thought-out (which is a good thing!), so it might be good to read through. For example, it states:

a number of experiments were performed which could be interpreted by classical mechanics only if it was assumed that electrons possessed a wave motion...

This statement reminded me that you may want to search these forums for discussions about the dichotomy between particle-like and wave-like behavior. It's the opinion of many that the distinction isn't so black and white as one might expect.
A: Probably only the collapse of the electron wave function, which occurs point-like in experiments. That means, electrons make point-like response on photographic plates, CRT screens and in other instruments, both in spatial and temporal sense. No wave description of this process has been built (yet).
Maybe also that several electrons make a many-particle wave function in higher dimensions, rather that just add up like classical waves. But this can be disputed.
