To debate on this subject, it is important to understand the properties and capabilities of the virus that can supposedly cause a zombie apocalypse.
Let us the consider the case of The Walking Dead show, as an example;
1. The virus that brings the dead back to life is not dumb but intelligent to a certain degree. Hints indicate that it is bio-engineered.
2. Infected bodies don't decay very fast because of the virus inside them which 'apparently' has some level of resistance against natural phenomenon that causes decomposition. Another way to look at this is that the virus apparently gives super-human strength or resistance to its host to keep it going against all odds for a while, minimizing damage from the prolonged and unhealthy exposure to the natural environment.
3. During reanimation process, basic instincts come back first. However, some additional brain functions also slowly come in to effect. Zombies (in TWD show) are capable of forming hordes/gangs instead of moving randomly and migrate/search for food sources with some level of cohesion. They are not very dumb.
4. Infection did not spread just through Zombie bites. Their were several others means. A decent hint is that the virus is inside the survivors too.
5. Keep in mind that the doctors and scientists who treat the infected are as much prone to getting infected in the process themselves. This can be supported through real life examples.
6. A major reason for the failure to contain the infection was the lack of cure. Some infections are simply too hard to fight effectively because they work at the DNA level.
These assumptions are not far-fetched when taking a peek at some real life examples.
Here is another interesting article.
I personally believe that The Thing like phenomenon is very realistic; a parasite that can impersonate its host and transform it in to a killing machine or reprogram it in a manner that the host serves as a vessel for spreading the infection. Such a parasite will be once again, intelligent to a certain level.