This work develops an envelope approach to time-dependent mechanism reliability defined in a period of time where a certain motion output is required. Since the envelope function of the motion error is not explicitly related to time, the time-dependent problem can be converted into a time-independent problem. The envelope function is approximated by piecewise hyperplanes. To find the expansion points for the hyperplanes, the approach linearizes the motion error at the means of random dimension variables, and this approximation is accurate because the tolerances of the dimension variables are small. The expansion points are found with the maximum probability density at the failure threshold. The time-dependent mechanism reliability is then estimated by a multivariable normal distribution at the expansion points. As an example, analytical equations are derived for a four-bar function generating mechanism. The numerical example shows the significant accuracy improvement.