This paper considered the comparison of some tests for assessing the overall homogeneity of Kaplan-Meier survival curves under low and high censoring rates when the curves are disjointed towards the end. The performances of these tests were measured by their statistical powers. Monte Carlo simulation study was conducted to evaluate and numerically compare the relative performances of Log-rank,Wilcoxon, Tarone-Ware, Peto-Peto, Modified Peto-Peto, the Fleming-Harrington (1,1), and the Babalola-Adeleke tests. The result obtained shows that the Babalola-Adeleke and Fleming-Harrington (1,1) tests have more robust performances than the other five popular tests with relatively high power in detecting differences when the censoring rates in the groups are both low and high. The highest overall average powers under low and high censoring rates were produced by Babalola-Adeleke and Fleming-Harrington (1,1) tests respectively. Hence, these two tests are the most suitable tests for diagnosing homogeneity of survival curves under these conditions.