We investigate the problem of $N$ identical bosons that are coupled to an impurity particle with infinite mass. For noninteracting bosons, we show that a dynamical impurity-boson interaction, mediated by a closed-channel dimer, can induce an effective boson-boson repulsion which strongly modifies the bound states consisting of the impurity and $N$ bosons. In particular, we demonstrate the existence of two universal “multibody” resonances, where all multibody bound states involving any $N$ emerge and disappear. The first multibody resonance corresponds to infinite impurity-boson scattering length, $a\to +\infty$, while the second corresponds to the critical scattering length $a^∗>0$ beyond which the trimer ($N=2$ bound state) ceases to exist. Crucially, we show that the existence of $a^∗$ ensures that the ground-state energy in the multibody bound-state region, $\infty>a>a^∗$, is bounded from below, with a bound that is independent of $N$. Thus, even though the impurity can support multibody bound states, they become increasingly fragile beyond the dimer state. This has implications for the nature of the Bose polaron currently being studied in cold-atom experiments.