Obtaining initial conditions and parameterizations leading to a model consistent with available measurements or safety specifications is important for many applications. Examples include model (in-)validation, prediction, fault diagnosis, and controller design. We present an approach to determine inner- and outer-approximations of the set containing all consistent initial conditions/parameterizations for nonlinear continuous-time systems. These approximations are found by occupation measures that encode the system dynamics and measurements, and give rise to an infinite-dimensional linear program. We exploit the flexibility and linearity of the decision problem to incorporate uncertain-but-bounded and pointwise-in-time state and output constraints, a feature which was not addressed in previous works. The infinite-dimensional linear program is relaxed by a hierarchy of LMI problems that provide certificates in case no consistent initial condition/parameterization exists. Furthermore, the applied LMI relaxation guarantees that the approximations converge (almost uniformly) to the true consistent set. We illustrate the approach with a biochemical reaction network involving unknown initial conditions and parameters.