A Unity ID allows you to buy and/or subscribe to Unity products and services, shop in the Asset Store and participate in the Unity community.
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I am not sure how what Tautvydas wrote solves the problem but it solved my problem as well. Many thanks!
It is a bug in Unity 5.3.x. I uninstalled Unity 5.3.1 and installed Unity 5.2.3. It solved the problem 100%.