Carbon-fiber-reinforced polymer (CFRP) has gradually become a new material to replace traditional steel due to its outstanding advantages. Because of its poor transverse stress performance, there is a reduction effect on the tensile strength in the bending state. To study the mechanical properties of CFRP tendons subjected to combined tension and bending at the saddle of a suspension bridge, a series of bond-type anchorages were made. Specimens with different diameters of CFRP tendons were tensioned on the device with different bending radius saddles. The test results revealed that the tensile properties were significantly affected by the severity of the bending of the CFRP tendons, including the failure mode, fracture force, and stress distribution. The highest reduction in fracture force was found at the bending radius of 3 m, of up to 38.05%. Furthermore, the tensile properties were also found to be influenced by the diameter of CFRP tendons. It was found that increasing the bending radius was more conducive to improving the performance of CFRP tendons with a smaller diameter. When the bending radius increased from 3 to 12 m, the efficiency coefficient (the ratio of the fracture force to the ultimate force) of D8, D10, and D14 increased by 11.21%, 7.74%, and 2.26%, respectively. Decreasing the bending radius leads to unevenness of the stress distribution and increasing the diameter of the CFRP tendon leads to brittleness and difficulties in anchoring, thus resulting in the decrease in the efficiency coefficient. In addition, the ratio of the bending radius to the tendon diameter was less than 2.4, the efficiency coefficient of the specimen was less than 80%, and the specimen mostly suffered shear failure. Furthermore, the finite element (FE) models validated by the test results were used to reveal the stress state and study the effect of contact friction on the properties of CFRP tendons. The FE results show that the CFRP tendons with a smaller bending radius presented higher shear stress concentrations. As the contact friction increased, the load-bearing capacity of CFRP tendons decreased significantly.