We present the results of our high-resolution spectroscopic follow-up of 42 planetary transit candidates in Carina from the OGLE survey . This follow-up has already allowed the discovery of three new transiting exoplanets , OGLE-TR-111 , 113 and 132 , presented in earlier Letters ( ) . Here we analyse the data for the remaining 39 candidates . The radial velocity data show that most of them are eclipsing binaries , in very varied configurations . Precise radial velocity orbits were derived for 15 binaries , revealing 9 transits of small stars ( generally M-dwarfs ) in front of F-G dwarfs , 1 grazing equal-mass eclipsing binary , 4 triple and 1 quadruple systems . A remaining 14 systems appear binary , but the exact orbit is uncertain or was not determined . 2 objects do not show any radial velocity variations in phase with the transit signal , and 6 do not possess spectral lines strong enough for a reliable cross-correlation function to be measured . Among these last two categories , up to 6 objects are suspected false positives of the photometric transit detection . Finally 2 objects are unsolved cases that deserve further observations . Our study illustrates the wide variety of cases that can mimic photometric planetary transits , and the importance of spectroscopic follow-up . Multi-fiber capacities and an optimized follow-up strategy , which we present here , can help deal with the high number of candidates that are likely to turn up in the near future . An important by-product of this study is the determination of exact masses and radii for six very low-mass stars , including two at the very edge of the stellar domain , OGLE-TR-106 ( M = 0.116 \pm 0.021 { \mathrm { M } } _ { \odot } ) and OGLE-TR-122 ( M = 0.089 \pm 0.007 { \mathrm { M } } _ { \odot } ) . The radius of these objects is consistent with theoretical expectations . Two further objects , OGLE-TR-123 and OGLE-TR-129 , may harbour transiting companions near the brown-dwarf/stellar limit ( M \simeq 0.07 { \mathrm { M } } _ { \odot } ) , whose confirmation requires further high-resolution spectroscopic monitoring . No transiting massive planets ( M = 2 - 10 { \mathrm { M } } _ { \mathrm { J } } ) were detected , confirming the rarity of such systems at short period as indicated by Doppler surveys . No light ( M < 0.5 { \mathrm { M } } _ { \mathrm { J } } ) , large ( R > { \mathrm { R } } _ { \mathrm { J } } ) planets were found either , indicating that ” hot Saturns ” generally have smaller radii than hot Jupiters . Three short period binaries with a M-dwarf companion show definite orbital eccentricities , with periods ranging from 5.3 to 9.2 days . This confirms theoretical indications that orbital circularisation in close binaries is less efficient for smaller companion masses . We also discuss the implications of our results for the statistical interpretation of the OGLE planetary transit survey in Carina in terms of planet frequency and detection efficiency . We find that the actual transit detection threshold is considerably higher than expected from simple estimates , and very strongly favours the detection of planets with periods shorter than about 2 days . The apparent contradition between the results of the OGLE transit survey and Doppler surveys can be resolved when this detection bias is taken into account .