No, you have misunderstood. It is not natural to have high morning Bg levels !
I will try to explain. It just means if your levels drop low in the night the DP effect kicks in and elevates your levels automatically. The trick is to make sure that your pre bed level is not too low as this will definitely trigger the DP. I don't often experience it because my pre bed level is usually around 7 - 8 mmol/l. This is enough for me to get up, test, then have breakfast with little or no DP effect.
If however, I get up fail to test, fail to eat for a while or exercised then if it hadn't already kicked in it will then, so my levels would go up. Without having eaten a thing ! The high level AM would then as you have found carry over throughout the day, unless you try to counteract it by some exercise, drinking plenty of fluids, water etc. If you are also eating loads of carbs then it just gets worse and worse.
What exactly do you have for your breakfast ?