Can a mistake of a single air-traffic controller cause a collision between planes?
I haven't seen the Breaking Bad episode, but in my experience any aviation related scenes in films and series are usually stretching reality beyond the breaking point. 

But it is possible that a single mistake can cause an accident. The probability of that happening is extremely remote, though.

Typically there are a number of safety barriers (also called defensive layers) in the system(1) that prevent any single mistake from developing into a collision. However, every barrier has a small probability of not catching the problem (e.g. due to it not being perfect or being in state of maintenance / failure) so eventually a crash could result from a single mistake. 

This is often called the Swiss Cheese Model (originally from Reason, J. (1990) Human Error. Cambridge: University Press, Cambridge.)




When an accident happens, sometimes the phrase "all the holes in the Swiss cheese lined up" is heard, and you now know why. 

While the first hole might be the mistake of a controller, the other holes may not be mistakes but are certainly rare circumstances and flaws in the system.  Accident investigation focuses on identifying these contributing factors.

In the &#220;berlingen accident mentioned by Portree Kid, two immediate causes have been identified(2):


  
  The imminent separation infringement was not noticed by ATC in time. The instruction for the TU154M to descend was given at a time when the prescribed separation to the B757-200 could not be ensured any more.
  The TU154M crew followed the ATC instruction to descend and continued to do so even after TCAS advised them to climb. This manoeuvre was performed contrary to the generated TCAS RA.
  


The first cause can be seen as a mistake by the controller, however that mistake could only happen because of a number of contributing factors (systemic causes):


  
  Management and quality assurance of the air navigation service company did not ensure that during the night all open workstations
  were continuously staffed by controllers.
  Management and quality assurance of the air navigation service company tolerated for years that during times of low traffic flow at night only one controller worked and the other one retired to rest.
  


In addition to the above, the Short Term Conflict Alerting (STCA) system did not display conflicts to the air traffic controller because of maintenance (not listed as a cause in the report, but discussed in the text).

The second cause is not a mistake by the crew of the TU154M, but a latent failure in the way ACAS/TCAS II was introduced. Never did anyone give guidance on what to do if the controller and the TCAS gave conflicting instructions. 

In the report this is identified as a systemic cause:


  
  The integration of ACAS/TCAS II into the system aviation was insufficient and did not correspond in all points with the system philosophy. The regulations concerning ACAS/TCAS published by ICAO and as a result the regulations of national aviation authorities, operational and procedural instructions of the TCAS manufacturer and the operators were not standardised, incomplete and partially contradictory.
  


So while the accident was caused by a single mistake from the controller, there were also many flaws in the system, which all together allowed the mistake to develop into an accident. 

(1) By system I mean the aviation system as a whole, not only the air traffic control (computer) systems.

(2) &#220;berlingen accident report