We study the optical reflectivity of real three-dimensional (3D) photonic band-gap crystals with increasing thickness. The crystals consist of GaAs plates with nanorod arrays that are assembled by an advanced stacking method into high-quality 3D woodpile structures. We observe intense and broad reflectivity peaks with stop bands that correspond to a broad gap in the photonic band structures. The maximum reflectivity quickly reaches high values, even for a few crystal layers. Remarkably, the bandwidth of the stop bands hardly decreases with increasing crystal thickness, in good agreement with finite-difference time domain (FDTD) simulations. This behavior differs remarkably from the large changes observed earlier in weakly interacting 3D photonic crystals. The nearly constant bandwidth and high reflectivity are rationalized by multiple Bragg interference that occurs in strongly interacting photonic band-gap crystals, whereby the incident light scatters from multiple reciprocal lattice vectors simultaneously, in particular, from oblique ones that are parallel to a longer crystal dimension and thus experience hardly any finite-size effects. Our insights have favorable consequences for the application of 3D photonic band-gap crystals, notably since even thin structures reveal the full band-gap functionality, including devices that shield quantum bits from vacuum fluctuations.