In reconstructing exposure histories needed to calculate cumulative exposures, gaps often occur. Our investigation was motivated by case-control studies of residential radon exposure and lung cancer, where half or more of the targeted homes may not be measurable. Investigators have adopted various schemes for imputing exposures for such gaps. We first undertook simulations to assess the performance of five such methods under an excess relative risk model, in the presence of random missingness and under assumed independence among the true exposure levels for different epochs of exposure (houses). Assuming no other source of measurement error, one of the methods performed without bias and with coverage of nominally 95% confidence intervals that was close to 95%. This method assigns to the missing residences the arithmetic mean across all measured control residences. We show that its good properties can be explained by the fact that this approach produces approximate "Berkson errors." To take advantage of predictive information that might exist about the missing epochs of exposure, one might prefer to carry out the imputations within strata. In further simulations, we asked whether the method would still perform well if imputations were carried out within many strata. It does, and much of the lost statistical power/precision can be recovered if the stratification system is moderately predictive of the missing exposures. Thus, observed control mean imputation provides a way to impute missing exposures without corrupting the study's validity; and stratifying the imputations can enhance precision. The technique is applicable in other settings where exposure histories contain gaps.