We are monitoring 27 galaxies within 10 Mpc using the Large Binocular Telescope to search for failed supernovae (SNe), massive stars that collapse to form a black hole without an SN explosion. We present the results from the first 4 yr of survey data, during which these galaxies were observed to produce three successful core-collapse SNe. We search for stars that have 'vanished' over the course of our survey, by examining all stars showing a decrease in luminosity of ...L... greater than or equal to 10 ...L... from the first to the last observation. We also search for the low luminosity, long duration transients predicted by Lovegrove & Woosley (2013) for failed explosions of red supergiants. After analysing the first 4 yr of data in this first direct search for failed SNe, we are left with one candidate requiring further study. This candidate has an estimated mass of 18-25 M..., a mass range likely associated with failed SNe and, if real, implies that failed SN represents a median fraction of f ... 0.30 of core collapses, with symmetric 90 per cent confidence limits of 0.07 less than or equal to f less than or equal to 0.62. If follow up data eliminate this candidate, we find an upper limit on the fraction of core collapses leading to a failed SN of f < 0.40 at 90 per cent confidence. As the duration of the survey continues to increase, it will begin to constrain the f ... 10-30 per cent failure rates needed to explain the deficit of massive SN progenitors and the observed black hole mass function. (ProQuest: ... denotes formulae/symbols omitted.)