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Structural identification through finite element model updating has gained increased importance as an applied experimental technique for performance-based structural assessment and health monitoring. However, practical challenges associated with computability, feasibility, and uniqueness present in the structured nonlinear inverse eigenvalue problem develop as a result of the necessary use of partially described and incompletely measured mode shapes. As an alternative to direct methods and optimization-based approaches, this paper proposes a new paradigm for model updating that is based on formulating the structured inverse eigenvalue problem as a Constraint Satisfaction Problem. Interval arithmetic and contractor programming are introduced as a means for generating feasible solutions to a structured inverse eigenvalue problem within a bounded parameter search space. This framework offers the ability to solve under-determined and non-unique inverse problems as well as accommodate measurement uncertainty through relaxation of constraint equations. These abilities address key challenges in quantifying uncertainty in parameter estimates obtained through structural identification and enable the exploration of alternative solutions to the global minimum that may better reflect the true physical properties of the structure. These capabilities are first demonstrated using synthetic data from a numerical mass-spring model and then extended to experimental data from a laboratory shear building model. Lastly, the methodology is contrasted with probabilistic model updating to highlight the advantages and unique capabilities offered by the methodology in addressing the effects of measurement uncertainty on the parameter estimation.