The basic assumptions of the Cox proportional hazards regression model are rarely questioned. This study addresses whether hazard ratio, i.e., relative risk (RR), estimates using the Cox model are biased when these assumptions are violated. We investigated also the dependence of RR estimates on temporal exposure characteristics, and how inadequate control for a strong, time-dependent confounder affects RRs for a modest, correlated risk factor. In a realistic cohort of 500,000 adults constructed using the National Cancer Institute Smoking History Generator, we used the Cox model with increasing control of smoking to examine the impact on RRs for smoking and a correlated covariate X. The smoking-associated RR was strongly modified by age. Pack-years of smoking did not sufficiently control for its effects; simultaneous control for effect modification by age and time-dependent cumulative exposure, exposure duration, and time since cessation improved model fit. Even then, residual confounding was evident in RR estimates for covariate X, for which spurious RRs ranged from 0.980 to 1.017 per unit increase. Use of the Cox model to control for a time-dependent strong risk factor yields unreliable RR estimates unless detailed, time-varying information is incorporated in analyses. Notwithstanding, residual confounding may bias estimated RRs for a modest risk factor.