We investigate electron paramagnetic resonance spectra of bismuth-doped silicon, at intermediate magnetic fields B≃0.1-0.6 T, theoretically and experimentally (with 9.7 GHz X-band spectra). We identify a previously unexplored regime of "cancellation resonances," where a component of the hyperfine coupling is resonant with the external field. We show that this regime has experimentally accessible consequences for quantum information applications, such as reduction of decoherence, fast manipulation of the coupled electron-nuclear qubits, and spectral line narrowing.