Precise determinations of the masses of the W boson and of the top quark could stringently test the radiative structure of the Standard Model (SM) or provide evidence for new physics. We analyze the excellent prospects at a muon collider for measuring M-W and m(t) in the W+W- and ZN threshold regions. With an integrated luminosity of 10 (100) fb(-1), the W-boson mass could be measured to a I,recision of 20 (6) MeV, provided that theoretical and experimental systematics are understood. A measurement of Delta M-W = 6 MeV would constrain the mass of a similar to 100 GeV Higgs to about +/-10 GeV. We demonstrate that a measurement at future colliders of the Bjorken process With an integrated luminosity of 100 fb(-1) it is possible to measure the Standard Model Higgs mass to within 45 MeV at a mu(+)mu(-) collider for m(h) = 100 GeV.