This paper presents a robust, tube-based nonlinear model predictive controller for continuous-time systems with additive disturbances which cascades two sampled-data model predictive controllers: the first creates a desired path using nominal dynamics, and the second maintains the true state close to the nominal state by regulating a sliding variable designed on the error between the true and nominal states. The sampled-data model predictive approach permits easy incorporation of continuous-time sliding mode dynamics, allowing a dynamic boundary layer and tube design to be included. In this way, the control applied to the system capitalizes on the robustness properties of traditional sliding mode control (SMC) while incorporating system constraints. Stability analysis is presented in the context of input-to-state stability (ISS) for continuous-time systems. The proposed controller is implemented on two case studies, is compared to benchmark tube-based model predictive controllers, and is evaluated using average root-mean-square (RMS) values on the state and input variables, in addition to average integral square error (ISE) and integral absolute error (IAE) values on the position states. Results reveal that the proposed technique responds to higher levels of disturbance with significant increases in control effort, eliminates constraint violation by using of constrained SMC as the secondary controller, and maintains similar tracking performance to benchmark controllers at lower levels of control effort.