If the radiator is dirty inside enough and gets clogged to where it cannot allow the coolant to circulate properly or at all then yes the engine can overheat.
You can drain your vehicles radiator of coolant and then use a garden hose to flush the radiator out and see if any rust or other dirt comes out of the radiator.
Sometimes rust can get into the radiator especially if you do not use coolant and use water instead.
Water can be used short term to get somewhere but you should always use coolant to keep the engine from developing rust inside the engine and water pump and radiator.
You can also have your radiator and coolant system pressure tested and professionally cleaned out.
A radiator needs to circulate properly as well to allow the engine to stay cool enough and a clogged or dirty radiator or a bad radiator cap that is not sealing properly can cause the engine to overheat.