Field-programmable gate arrays (FPGAs) provide an opportunity to co-design applications with hardware accelerators, yet they remain difficult to program. High-level synthesis (HLS) tools promise to raise the level of abstraction by compiling C or C++ to accelerator designs. Repurposing legacy software languages, however, requires complex heuristics to map imperative code onto hardware structures. We find that the black-box heuristics in HLS can be unpredictable: changing parameters in the program that should improve performance can counterintuitively yield slower and larger designs. This paper proposes a type system that restricts HLS to programs that can predictably compile to hardware accelerators. The key idea is to model consumable hardware resources with a time-sensitive affine type system that prevents simultaneous uses of the same hardware structure. We implement the type system in Dahlia, a language that compiles to HLS C++, and show that it can reduce the size of HLS parameter spaces while accepting Pareto-optimal designs.