Dark matter that was once in thermal equilibrium with the Standard Model is generally prohibited from obtaining all of its mass from the electroweak or QCD phase transitions. This implies a new scale of physics and mediator particles needed to facilitate dark matter annihilations. In this work, we consider scenarios where thermal dark matter annihilates via scalar mediators that are colored and/or electrically charged. We show how partial wave unitarity places upper bounds on the masses and couplings on both the dark matter and mediators. To do this, we employ effective field theories with dark matter as well as three flavors of sleptons or squarks with minimum flavor violation. For Dirac (Majorana) dark matter that annihilates via mediators charged as left-handed sleptons, we find an upper bound around 45 TeV (7 TeV) for both the mediator and dark matter masses, respectively. These bounds vary as the square root of the number of colors times the number of flavors involved. Therefore the bounds diminish by root two for right handed selectrons. The bounds increase by root three and root six for right and left handed squarks, respectively. Finally, because of the interest in natural models, we also focus on an effective field theory with only stops. We find an upper bound around 32 TeV (5 TeV) for both the Dirac (Majorana) dark matter and stop masses. In comparison to traditional naturalness arguments, the stop bound gives a firmer, alternative expectation on when new physics will appear. Similar to naturalness, all of the bounds quoted above are valid outside of a defined fine-tuned regions where the dark matter can co-annihilate. The bounds in this region of parameter space can exceed the well-known bounds from Griest and Kamionkowski. We briefly describe the impact on planned and existing direct detection experiments and colliders. Show Partial Abstract