Ferromagnetic inverse opal-like structures can be considered as ordered three-dimensional network of nanoislands connected to each other by elongated links. These systems can serve as a playground for studying 3D nanomagnetism. At the same time the structure period (700 nm) and sample size are large enough to apply convenient integral and surface sensitive experimental techniques, but not small enough to use methods that are able to recover the full 3D magnetization distribution. In this regard, interpretation of experimental results should be carried out by means of simulations. We have studied the magnetic state of inverse opal-like structures by SQUID magnetometry and magnetic force microscopy in an external magnetic field. Results have been compared with micromagnetic simulations. We have found that the experimental data can be well interpreted in the frame of the spin-ice model suggested earlier. In particular the transversal magnetization predicted by this model has been found in fields applied along the FCC axis. Its field dependence is in agreement with the results of the calculations.