If you can have uncertainty in momentum, then wouldn't you have uncertainty in mass and velocity?
Why can't mass be uncertain?
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Uncertainty is a property of observables. Mass is not normally taken to be an observable, so it does not obey uncertainty relations.
Why isn't mass an observable? There is a superselection rule that forbids it in the presence of reasonable symmetry assumptions. See the discussion here for more.
EDIT: In "true" relativistic QFT one wouldn't even talk about "mass" but "mass-energy", and Bargmann's superselection rule doesn't hold. In that context mass-energy is a well-defined observable, and it obeys an energy-time uncertainty relation.
Based on the simple definition of the Heisenberg uncertainty principal, uncertainty is an inverse ratio between location and velocity, so if you're willing to know very little about location, then you can measure velocity with accuracy.
Now, on the quantum level, particles can do strange things like borrow energy from the future so, there's probably a small and perhaps temporary level of uncertainty always, but beyond that, the uncertainty principal allows you to measure mass, velocity or momentum precisely so long as you don't care about location.