To reproduce: 
1. Add RadCheckedDropDownList with few items 
2. Subscribe to the ItemCheckedChanged event
3. In handler of event add the following code snippet: 
void radCheckedDropDownList1_ItemCheckedChanged(object sender, RadCheckedListDataItemEventArgs e)
{
    this.radCheckedDropDownList1.Text = string.Format("Count {0}", radCheckedDropDownList1.CheckedItems.Count.ToString());
}
4. Open the drop down popup and you will see that can not check/uncheck any item. 


Workaround: 
1. Subscribe to the TextChanged event of AutoCompleteEditableAreaElement and set the Text property: 

this.radCheckedDropDownList1.CheckedDropDownListElement.AutoCompleteEditableAreaElement.AutoCompleteTextBox.TextChanged += AutoCompleteTextBox_TextChanged;
void AutoCompleteTextBox_TextChanged(object sender, EventArgs e)
{
    this.radCheckedDropDownList1.Text = string.Format("Count {0}", radCheckedDropDownList1.CheckedItems.Count.ToString());
}