A careful analysis of the backward recurrence algorithm for evaluating approximants of continued fractions provides rigorous bounds for the accumulated relative error due to rounding. Such errors are produced by machine operations which carry only a fixed number v of significant digits in the computations. The resulting error bounds are expressed in terms of the machine parameter v. The derivation uses a basic assumption about continued fractions, which has played a fundamental role in developing convergence criteria. Hence, its appearance in the present context is quite natural. For illustration, the new error bounds are applied to two large classes of continued fractions, which subsume many expansions of special functions of physics and engineering, including those represented by Stieltjes fractions. In many cases, the results insure numerical stability of the backward recurrence algorithm.