No,it doesn't have to be less.
Work Step by Step
If you have a dataset with all equal numbers the third quartile and the maximum will all be equal. For example, if you have 24,24,24,24,24,24.The minimum will be 24, The first quartile will be 24, The median will be 24, The third quartile will be 24, The maximum will be 24. Therefore, all the values will be same so it doesn't have to be less