When analyzing single-molecule data, a low-dimensional set of system observables typically serve as the observational data. We calibrate stochastic dynamical models from time series that record such observables. Numerical techniques for quantifying noise from multiple time-scales in a single trajectory, including experimental instrument and inherent thermal noise, are demonstrated. The techniques are applied to study time series coming from both simulations and experiments associated with the nonequilibrium mechanical unfolding of titin's I27 domain. The estimated models can be used for several purposes: (1) detect dynamical signatures of "rare events" by analyzing the effective diffusion and force as a function of the monitored observable, (2) quantify the influence that conformational degrees of freedom, which are typically difficult to directly monitor experimentally, have on the dynamics of the monitored observable, (3) quantitatively compare the inherent thermal noise to other noise sources, e.g. instrument noise, variation induced by conformational heterogeneity, etc., (4) simulate random quantities associated with repeated experiments, (5) apply pathwise, i.e. trajectory-wise, hypothesis tests to assess the goodness-of-fit of the models and even detect conformational transitions in noisy signals. These items are all illustrated with several examples.