A failsafe network design recovering from the stressed condition against a massive supply disruption is generally useful for various applications. Water flow in plants under a tension is inherently vulnerable to an embolism, a water supply cut off, causing a death. However, the function of the network structures of leaf veins and xylem stems effectively reduces the embolism-induced failure. In this study, water transport in plants under the pressurized conditions compared to the normal physiological conditions is observed by X-ray imaing. By examining embolism-induced water supply limits in the architecturally diverse leaf and stem networks, a progressive hydraulic rule has been found: the limited flows in the selected parts of the network structures against a total fail. For a scientific explanation on nanoscale water flow dynamics occurring in plants, temporal meniscus development in the nanomembrane model system is investigated. The pressure-driven hydrodynamic transport phenomena can be explained to follow network dynamics of the modified imbibition typically occuring in nanostrutcures. This study contributes to a variety of design technologies of networked materials against the spread of flow damages under the stressed conditions.