Context. Long-baseline interferometry is an important technique to spatially resolve binary or multiple systems in close orbits. By combining several telescopes together and spectrally dispersing the light, it is possible to detect faint components around bright stars in a few hours of observations. Aims. We provide a rigorous and detailed method to search for high-contrast companions around stars, determine the detection level, and estimate the dynamic range from interferometric observations. Methods. We developed the code CANDID (Companion Analysis and Non-Detection in Interferometric Data), a set of Python tools that allows us to search systematically for point-source, high-contrast companions and estimate the detection limit using all interferometric observables, i.e., the squared visibilities, closure phases and bispectrum amplitudes. The search procedure is made on a N × N grid of fit, whose minimum needed resolution is estimated a posteriori. It includes a tool to estimate the detection level of the companion in the number of sigmas. The code CANDID also incorporates a robust method to set a 3σ detection limit on the flux ratio, which is based on an analytical injection of a fake companion at each point in the grid. Our injection method also allows us to analytically remove a detected component to 1) search for a second companion; and 2) set an unbiased detection limit. Results. We used CANDID to search for the companions around the binary Cepheids V1334 Cyg, AX Cir, RT Aur, AW Per, SU Cas, and T Vul. First, we showed that our previous discoveries of the components orbiting V1334 Cyg and AX Cir were detected at >25σ and >13σ, respectively. The astrometric positions and flux ratios provided by CANDID for these two stars are in good agreement with our previously published values. The companion around AW Per is detected at more than 15σ with a flux ratio of f = 1.22 ± 0.30%, and it is located at ρ = 32.16 ± 0.29 mas and PA = 67.1 ± 0.3• . We made a possible detection of the companion orbiting RT Aur with f = 0.22 ± 0.11%, and at ρ = 2.10 ± 0.23 mas and PA = −136 ± 6• . It was detected at 3.8σ using the closure phases only, and so more observations are needed to confirm the dectection. No companions were detected around SU Cas and T Vul. We also set the detection limit for possible undetected companions around these stars. We found that there is no companion with a spectral type earlier than B7V, A5V, F0V, B9V, A0V, and B9V orbiting the Cepheids V1334 Cyg, AX Cir, RT Aur, AW Per, SU Cas, and T Vul, respectively. This work also demonstrates the capabilities of the MIRC and PIONIER instruments, which can reach a dynamic range of 1:200, depending on the angular distance of the companion and the (u, v) plane coverage. In the future, we plan to work on improving the sensitivity limits for realistic data through better handling of the correlations.