Dynamic force spectroscopy provides insight into the structure and dynamics of interacting molecules or surfaces at a molecular level by applying mechanical forces in a controllable way. The use of atomic force microscopy (AFM) in single-molecule pulling experiments allows studying forced escape, detachment, dissociation events associated with an increasing force field, and determining bonds’ strength. Here, we consider the rupture force spectra in a poly(ethyleneimine) (PEI) model system deposited on a silica surface displaying multiple dissociation events in contact with the AFM probe and analyze statistical properties of rupture forces derived from the multiple repeated experiments. We find that the obtained histograms are overdispersed. The most probable rupture force decreases with the increase of pulling velocity being of poor agreement with models based on Kramers theory describing the crossing event through a single energy barrier. The experimental data of multi-breakdown events are shown to be well fitted with the Weibull distribution, which stems from generally nonexponential molecular relaxation under the action of applied stress. We also provide a numerical study of a simple microscopic polymer chain model with multiple bonds with the surface, as well as a stochastic PEI model. The obtained results support our experimental and theoretical findings qualitatively.