Fiber-reinforced thermoset composites are a class of materials that address the arising needs from the aerospace and hypersonic industries for high specific strength, temperature-resistant structural materials. Among the high-temperature resistant thermoset categories, phenolic triazine (PT) cyanate esters stand out thanks to their inherent high degradation temperature, glass transition temperature, and mechanical strength. Despite the outstanding properties of these thermosets, the performance of carbon fiber composites using PT cyanate esters as matrices has not been thoroughly characterized. This work evaluated PT and carbon fiber composites’ compressive properties and failure mechanisms with different fiber arrangements. A PT resin with both plain weave (PW) and non-crimped unidirectional (UD) carbon fiber mats was analyzed in this research. Highly loaded thermoset composites were obtained using process temperatures not exceeding 260 °C, and the composites proved to retain compressive strength at temperatures beyond 300 °C. Compressive testing revealed that PT composites retained compressive strength values of 50.4% of room temperature for UD composites and 61.4% for PW composites. Post-compressive failure observations of the gage section revealed that the mechanisms for failure evolved with temperature from brittle, delamination-dominant failure to shear-like failure promoted by the plastic failure of the matrix. This study demonstrated that PT composites are a good candidate for structural applications in harsh environments.