Does a conservation law limit position measurements?

The demonstrations of Wigner and others, that observables which do not commute with additive conserved quantities cannot be measured precisely, are reexamined. A proposed new formulation of the claim is shown to be valid for observables with a continuous spectrum whenever the conserved quantities are bounded. However, a countermodel is constructed, and it suggests that the position can be measured as precisely as the momentum even though the measuring interaction conserves the total linear momentum.