In this article, the problem of model identification for linear systems is considered, using a finite set of sampled data affected by a bounded measurement noise, with unknown bound. The objective is to identify one-step-ahead models and their accuracy in terms of worst-case simulation error bounds. To do so, the set membership identification framework is exploited. Theoretical results are derived, allowing one to estimate the noise bound and system decay rate. Then, these quantities and the data are employed to define the feasible parameter set (FPS), which contains all possible models compatible with the available information. Here, the estimated decay rate is used to refine the standard FPS formulation, by adding constraints that enforce the desired converging behavior of the models' impulse response. Moreover, guaranteed simulation error bounds for an infinite future horizon are derived, improving over recent results pertaining to finite simulation horizon only. These bounds are the basis for a result and method to guarantee asymptotic stability of the identified model. Finally, the desired one-step-ahead model is identified by means of numerical optimization, and the related simulation error bounds are evaluated. Both input-output and state-space model structures are addressed. The approach is showcased on a numerical example and on real-world experimental data of the roll rate dynamics of an autonomous glider.