The unique properties of hybrid organic-inorganic perovskites (HOIPs) promise to open doors to next-generation flexible opto-electronic devices. Before such advances are realized, a fundamental understanding of the mechanical properties of HOIPs is required. Here, we combine ab-initio density functional theory (DFT) modeling with a diverse set of experiments to study the elastic properties of (quasi)2D HOIPs. Specifically, we focus on (quasi)2D single crystals of phenethylammonium methylammonium lead iodide, (PEA)2PbI4(MAPbI3)n-1, and their 3D counterpart, MAPbI3. We used nano-indentation (both Hertzian and Oliver-Pharr analyses) in combination with elastic buckling instability experiments to establish the out-of-plane and in-plane elastic moduli. The effect of van der Waals (vdW) forces, different interlayer interactions and finite temperature are combined with DFT calculation to accurately model the system. Our results reveal a non-monotonic dependence, of both the in-plane and out-of plane elastic moduli on the number of inorganic layers (n) rationalized by first-principles calculations. We discuss how the presence of defects in as-grown crystals and macroscopic interlayer deformations affect the mechanical response of (quasi)2D HOIPs. Comparing the in- and out-of-plane experimental results with theory reveals that perturbations to the covalent and ionic bonds (which hold a 2D-layer together) is responsible for the relative out-of-plane stiffness of these materials. In contrast, we conjecture that the in-plane softness originates from macroscopic or mesoscopic motions between 2D-layers during buckling experiments. Additionally, we learn how dispersion and π interactions in organic bilayers can have a determining role on the elastic response of the materials, especially in the out-of-plane direction. The understanding gained by comparing ab-initio and experimental techniques paves the way for rational design of layered HOIPs with mechanical properties favorable for strain-intensive applications. Combined with filters for other favorable criteria e.g., thermal or moisture stability, one can systematically screen viable (quasi)2D HOIPs for a variety of flexible optoelectronic applications.