The problem of statistically bounding the response of an engineering structure with random boundary conditions is addressed across the entire frequency range: from the low, through the mid, to the high frequency region. Extreme-value-based bounding of both the FRF and the energy density response is examined for a rectangular linear plate with harmonic point forcing. The proposed extreme-value (EV) approach, previously tested only in the low frequency region for uncoupled and acoustically-coupled uncertain structures, is examined here in the mid and high frequency regions in addition to testing at low frequencies. EV-based bounding uses an asymptotic threshold exceedance model of Type-I, to extrapolate the m-observational return period to an arbitrarily-large batch of structures. It does this by repeatedly calibrating the threshold model at discrete frequencies using a small sample of response data generated by Monte Carlo simulation or measurement. Here the discrete singular convolution (DSC) method – a transfrequency computation approach for deterministic vibration - is used to generate Monte Carlo samples. The accuracy of the DSC method is first verified (i) in terms of the spatial distribution of total energy density and (ii) across the frequency range, by comparison with a mode superposition method and Statistical Energy Analysis (SEA). EV-based bound extrapolations of the receptance FRF and total energy density are then compared with: (i) directly-estimated bounds using a full set of Monte Carlo simulations and (ii) with total mean energy levels obtained with SEA. This paper shows that for a rectangular plate structure with random boundary conditions, EV-based statistical bounding of both the FRF and total energy density response is generally applicable across the entire frequency range.