Context. Studies of transiting extrasolar planets are of key importance for understanding the nature of planets outside our solar system because their masses, diameters, and bulk densities can be measured. An important part of transit-search programmes is the removal of false-positives. In the case of the CoRoT space mission, the majority of the false-positives are removed by a detailed analysis of the light curves and by seeing-limited imaging in- and out-of-transit. However, the critical question is how many of the candidates that passed all these tests are false-positives. Such false-positives can be caused by eclipsing binaries, which are either related or unrelated to the targets. Aims: For our study we selected 25 CoRoT candidates that have already been screened against false-positives using detailed analysis of the light curves and seeing-limited imaging, which has transits that are between 0.7 and 0.05% deep. Our aim is to search for companion candidates that had not been recognized in previous observations. Methods: We observed 20 candidates with the adaptive optics imager NaCo and 18 with the high-resolution infrared spectrograph CRIRES. Results: We found previously unknown stars within 2'' of the targets in seven of the candidates. All of these are too faint and too close to the targets to have been previously detected with seeing-limited telescopes in the optical. Our study thus leads to the surprising results that if we remove all candidates excluded by the sophisticated analysis of the light-curve, as well as carrying out deep imaging with seeing-limited telescopes, still 28-35% of the remaining candidates are found to possess companions that are bright enough to be false-positives. Conclusions: Given that the companion candidates cluster around the targets and that the J - K colours are consistent with physical companions, we conclude that the companion candidates are more likely to be physical companions rather than unrelated field stars. Based on observations obtained at the European Southern Observatory at Paranal, Chile in programmes 282.C-5015A, 282.C-5015B, 282.C-5015C, 285.C-5045A, and 285.C-5045B, 086.C-0235A, 086.C-0235B, 088.C-0707A, 088.C-0707B, 090.C-0251A, 090.C-0251B, and 091.C-203(A).Appendices A and B are available in electronic form at http://www.aanda.org