Context. Only several doubly eclipsing quadruple stellar systems are known to date, and no dedicated effort to characterize population properties of these interesting objects has yet been made. Aims. Our first goal was to increase number of known doubly eclipsing systems such that the resulting dataset would allow us to study this category of objects via statistical means. In order to minimize biases, we used long-lasting, homogeneous, and well-documented photometric surveys. Second, a common problem of basically all known doubly eclipsing systems is the lack of proof that they constitute gravitationally bound quadruple system in the 2+2 architecture (as opposed to two unrelated binaries that are projected onto the same location in the sky by chance). When possible, we thus sought evidence for the relative motion of the two binaries. In that case, we tried to determine the relevant orbital periods and other parameters. Methods. We analysed photometric data for eclipsing binaries provided by the OGLE survey and we focused on the LMC fields. We found a large number of new doubly eclipsing systems (our discoveries are three times more numerous than the previously known cases in this dataset). In order to prove relative motion of the binaries about a common centre of mass, we made use of the fact that OGLE photometry covers several years. With a typical orbital period of days for the observed binaries, we sought eclipse time variations (ETVs) on the timescale comparable to a decade (this is the same method used for an archetype of the doubly eclipsing system, namely V994 Her). In the cases where we were able to detect the ETV period, the difference between the inner and outer periods in the quadruple system is large enough. This allows us to interpret ETVs primarily as the light-time effect, thus providing an interesting constraint on masses of the binaries. Results. In addition to significantly enlarging the database of known doubly eclipsing systems, we performed a thorough analysis of 72 cases. ETVs for 28 of them (39% of the studied cases) showed evidence of relative motion. Among these individual systems, we note OGLE BLG-ECL-145467, by far the most interesting case; it is bright (12.6 mag in I filter), consists of two detached binaries with periods of ≃3.3 d and ≃4.9 d (making it a candidate for a 3:2 resonant system) revolving about each other in only ≃1538 d. Distribution of the orbital period ratio P_A/P_B of binaries in 2+2 quadruples shows statistically significant excess at ≃1 and ≃1.5. The former is likely a natural statistical preference in weakly interacting systems with periods within the same range. The latter is thought to be evidence of a capture in the 3:2 mean motion resonance of the two binaries. This sets important constraints on evolutionary channels in these systems. Conclusions. The total number of doubly eclipsing systems increased to 146, more than 90% of which are at low declinations on the southern sky. This motivates us to use southern hemisphere facilities to further characterize these systems, and to seek possibilities to complement this dataset with northern sky systems.