The achievement of the low Gilbert damping parameter in spin dynamic modulation is attractive for spintronic devices with low energy consumption and high speed. Metallic ferromagnetic alloy Co-Fe-B is a possible candidate due to its high compatibility with spintronic technologies. Here, we report thickness-dependent damping and soft magnetism in Co-Fe-B films sandwiched between two non-magnetic layers with Co-Fe-B films up to 50 nm thick. A non-monotonic variation of Co-Fe-B film damping with thickness is observed, which is in contrast to previously reported monotonic trends. The minimum damping and the corresponding Co-Fe-B thickness vary significantly among the different non-magnetic layer series, indicating that the structure selection significantly alters the relative contributions of various damping mechanisms. Thus, we developed a quantitative method to distinguish intrinsic from extrinsic damping via ferromagnetic resonance measurements of thickness-dependent damping rather than the traditional numerical calculation method. By separating extrinsic and intrinsic damping, each mechanism affecting the total damping of Co-Fe-B films in sandwich structures is analyzed in detail. Our findings have revealed that the thickness-dependent damping measurement is an effective tool for quantitatively investigating different damping mechanisms. This investigation provides an understanding of underlying mechanisms and opens up avenues for achieving low damping in Co-Fe-B alloy film, which is beneficial for the applications in spintronic devices design and optimization.