Unit Test public methods related to a private collection

My class has a private lazy dictionary and some method to interact with it like this

public class MyClass : IExampleClass
{
    private Lazy<Dictionary<string, IMyObject>> myObjects = new Lazy<Dictionary<string, IMyObject>>();
    private Dictionary<string, IMyObject> MyObjects
    {
        get
        {
            return myObjects.Value;
        }
    }

    public class MyClass()
    {}

    public void RegisterObject(string name, IMyObject object)
    {
        logger.LogInfo("Executed RegisterObject etc.");

        if (String.IsNullOrEmpty(name) == false && object != null && MyObjects.ContainsKey(name) == false)
        {
             MyObjects.Add(name, object);
        }
        else
        {
             logger.LogWarning("Cannot register the object because etc.");
        }
    }

    public IMyObject GetObject(string name)
    {
        logger.LogInfo("Executed GetObject etc.");

        IMyObject obj = null;
        if (String.IsNullOrEmpty(name) == false)
        {
            Objects.TryGetValue(name, out obj);
        }

        return obj;
    }
}

This is something useful because sometimes happens that some of the inserted objects are null so i prevent any furter damage avoiding adding them. Now i'm facing a problem during the testing phase (i know, shame on me, i write code before test) because i'm forced to use both methods inside the same test:

[TestFixture]
public class MyClassTest 
{
    [Test]
    public void RegisterObject_WithNullNameAndNullObject_DoNothing()
    {
        //Arrange
        string inputName = null;
        IMyObject inputObject = null;
        IExampleClass sut = new MyClass();

        //Act
        sut.RegisterObject(inputName, inputObject);

        //Assert
        IMyObject actualValue = sut.GetObject(inputName);
        Assert.IsNull(actualValue);
    }

    ...
}

On top of that how can i handle the test where i'm trying to insert an element already in the dictionary?

    [Test]
    public void RegisterObject_AlreadyInserted_DoNothing()
    {
        //Arrange
        string inputName = "test";
        IMyObject inputObject = A.Dummy<IMyObject>();
        IExampleClass sut = new MyClass();

        // ??? 
        sut.RegisterObject(inputName, inputObject);
        // how can i effectively test the method if i'm using it more than once???

        //Act
        sut.RegisterObject(inputName, inputObject);

        //Assert
        IMyObject actualValue = sut.GetObject(inputName);

        // ???
        Assert.AreEqual(inputObject, actualValue);
        // What i've tested here? The fact that the dictionary contains
        // the object doesn't mean that the second call of RegisterObject
        // is ignored.
    }

So at end i'm asking:

1. Is correct to test RegisterObject calling GetObject inside it?


2. How can i arrange the dictionary if i have to test that the second insertion is ignored?

On a side note, i have 7 separate tests for RegisterObject:

• RegisterObject_WithNullNameAndNullObject_DoNothing()


• RegisterObject_WithNullNameAndValidObject_DoNothing()


• RegisterObject_WithEmptyNameAndNullObject_DoNothing()


• RegisterObject_WithEmptyNameAndValidObject_DoNothing()


• RegisterObject_WithValidNameAndNullObject_DoNothing()


• RegisterObject_WithValidNameAndValidObject_RegisterObject()


• RegisterObject_AlreadyInserted_DoNothing()

It's overkill? Too many test for this simple behavior?