Study on likability, weakness, and competence?
Thank you all for your responses. I found the answer independently - I was looking for the Pratfall Effect described in Elliot Aronson's 1966 paper "The effect of a pratfall on increasing interpersonal attractiveness" in Psychonomic Science. 

The paper's abstract reads:


  An experiment was performed which demonstrated that the attractiveness
  of a superior person is enhanced if he commits a clumsy blunder; the
  same blunder tends to decrease the the attractiveness of a mediocre
  person. We predicted these results by conjecturing that a superior
  person may be viewed as superhuman and, therefore, distant; a blunder
  tends to humanize him and, consequently, increases his attractiveness.


More information on the Pratfall Effect can be found here: Wikipedia, Psychology Archive 


Aronson, E., Willerman, B., &amp; Floyd, J. (1966). The effect of a pratfall on increasing interpersonal attractiveness. Psychonomic Science, 4(6), 227-228.