New results in ergodic theory show that averages of repeated measurements will typically diverge with probability one if there are random errors in the measurement of time. Since mean-square convergence of the averages is not so susceptible to these anomalies, we are led again to compare the mean and pointwise ergodic theorems and to reconsider efforts to determine properties of a stochastic process from the study of a generic sample path. There are also implications for models of time and the interaction between observer and observable.