We present new IRAM Plateau de Bure interferometer (PdBI) 1.3 mm continuum observations at ~1.5" resolution of 28 submillimeter galaxies (SMGs), previously discovered with the 870 μm bolometer LABOCA at the APEX telescope from the central 0.7 deg^2 of the COSMOS field. Nineteen out of the 28 LABOCA sources were detected with PdBI at a ≳3σ level of ≈1.4 mJy/beam. A combined analysis of this new sample with existing interferometrically identified SMGs in the COSMOS field yields the following results: i) ≳15%, and possibly up to ~40% of single-dish detected SMGs consist of multiple sources; ii) statistical analysis of multi-wavelength counterparts to single-dish SMGs shows that only ~50% have real radio or IR counterparts; iii) ~18% of interferometric SMGs have either no multi-wavelength counterpart or only a radio-counterpart; and iv) ~50–70% of z ≳ 3 SMGs have no radio counterparts (down to an rms of 7–12 μJy at 1.4 GHz). Using the exactinterferometric positions to identify the multi-wavelength counterparts allows us to determine accurate photometric redshifts for these sources. The redshift distributions of the combined and the individual 1.1 mm and 870 μm selected samples shows a higher mean and a broader width than those derived in previous studies. This study finds that on average brighter and/or mm- selected SMGs are located at higher redshifts, consistent with previous studies. The mean redshift for the 1.1 mm selected sample (z = 3.1 ± 0.4) is tentatively higher than that for the 870 μm selected sample ( = 2.6 ± 0.4). Based on our nearly complete sample of AzTEC 1.1 mm SMGs in a 0.15 deg^2 area, we infer a higher surface density of z ≳ 4 SMGs than predicted by current cosmological models. In summary, our findings imply that interferometric identifications at (sub-)millimeter wavelengths are crucial to build statistically complete and unbiased samples of SMGs.