The dipolar effect of neighboring protons that are not directly bonded to the carbon of interest on coupled carbon-13 relaxation in a simple organic molecule has been studied by comparing the relaxation behaviors of labeled carbon-13 in $Br13CH_2COOH$ with those in $BrCH_213COOH.$ Various pulse sequences, such as coupled inversion recovery pulse sequence, J-negative and J-positive pulse sequence, and nonselective and selective proton ${pi}pulse$ sequence, were employed to perform the required coupled spin relaxation experiments. To gain information on various spectral densities, including that of dipolar-CSA cross correlation, the experiments were performed on two different spectrometers, operating, respectively, at 50.31 and 125.51MHz for 13C. The magnitude of CH dipolar spectral densities for $BrCH_213COOH$ was found to be about 8% of those for $Br13CH_2COOH$, which means the effect due to the protons not directly bonded to the carbon of interest is small but not completely negligible.