Yes you can break your toe by stubbing your toe if you hit the toe or stub the toe hard enough on something.
However most times when you stub your toe you don't hit the object hard enough to cause any broken toe bones or any damage to the toe.
Most times stubbing your toe just causes some temporary pain which lasts for 1 to 2 hours or less but if the pain lasts longer than 1 day then you likely have broken a bone in your toe or broken your toe and if so you should see a doctor.
A stubbed toe should never hurt longer than a day and if it does then something is wrong.
If you just stubbed your toe then the pain from the stubbed toe should go away within a few hours to within the same day of stubbing the toe.
Whenever I've stubbed my toe the pain only lasted for 30 minutes to an hour depending on how hard I had stubbed my toe.
If you stubbed your toe and the pain persists longer than a few hours to more than a day or so then you might have actually broken your toe when you stubbed the toe.
Most times when you stub your toe it just creates some temporary pain without breaking the toe or doing any damage to the toe.
However if you happen to stub the toe hard enough then it is indeed possible to actually break the toe.
So if you stubbed your toe and the pain lasts for days then you likely broke the toe and should see a doctor about it.
Stubbed toes should not ever hurt for days and if the stubbed toe does hurt for days then you likely injured the toe worse than you thought.