You set the DisplayMemberPath and the SelectedValuePath to "Name", so I assume that you have a class PhoneBookEntry with a public property Name.

Have you set the DataContext to your ConnectionViewModel object?

I copied you code and made some minor modifications, and it seems to work fine. I can set the viewmodels PhoneBookEnty property and the selected item in the combobox changes, and I can change the selected item in the combobox and the view models PhoneBookEntry property is set correctly.

Here is my XAML content:

<Window x:Class="WpfApplication6.Window1"
        xmlns="http://schemas.microsoft.com/winfx/2006/xaml/presentation"
        xmlns:x="http://schemas.microsoft.com/winfx/2006/xaml"
        Title="Window1" Height="300" Width="300">
<Grid>
    <StackPanel>
        <Button Click="Button_Click">asdf</Button>
        <ComboBox ItemsSource="{Binding Path=PhonebookEntries}"
                  DisplayMemberPath="Name"
                  SelectedValuePath="Name"
                  SelectedValue="{Binding Path=PhonebookEntry}" />
    </StackPanel>
</Grid>
</Window>

And here is my code-behind:

namespace WpfApplication6
{

    /// <summary>
    /// Interaction logic for Window1.xaml
    /// </summary>
    public partial class Window1 : Window
    {
        public Window1()
        {
            InitializeComponent();
            ConnectionViewModel vm = new ConnectionViewModel();
            DataContext = vm;
        }

        private void Button_Click(object sender, RoutedEventArgs e)
        {
            ((ConnectionViewModel)DataContext).PhonebookEntry = "test";
        }
    }

    public class PhoneBookEntry
    {
        public string Name { get; set; }

        public PhoneBookEntry(string name)
        {
            Name = name;
        }

        public override string ToString()
        {
            return Name;
        }
    }

    public class ConnectionViewModel : INotifyPropertyChanged
    {
        public ConnectionViewModel()
        {
            IList<PhoneBookEntry> list = new List<PhoneBookEntry>();
            list.Add(new PhoneBookEntry("test"));
            list.Add(new PhoneBookEntry("test2"));
            _phonebookEntries = new CollectionView(list);
        }

        private readonly CollectionView _phonebookEntries;
        private string _phonebookEntry;

        public CollectionView PhonebookEntries
        {
            get { return _phonebookEntries; }
        }

        public string PhonebookEntry
        {
            get { return _phonebookEntry; }
            set
            {
                if (_phonebookEntry == value) return;
                _phonebookEntry = value;
                OnPropertyChanged("PhonebookEntry");
            }
        }

        private void OnPropertyChanged(string propertyName)
        {
            if (PropertyChanged != null)
                PropertyChanged(this, new PropertyChangedEventArgs(propertyName));
        }
        public event PropertyChangedEventHandler PropertyChanged;
    }
}

Edit: Geoffs second example does not seem to work, which seems a bit odd to me. If I change the PhonebookEntries property on the ConnectionViewModel to be of type ReadOnlyCollection, the TwoWay binding of the SelectedValue property on the combobox works fine.

Maybe there is an issue with the CollectionView? I noticed a warning in the output console:

System.Windows.Data Warning: 50 : Using CollectionView directly is not fully supported. The basic features work, although with some inefficiencies, but advanced features may encounter known bugs. Consider using a derived class to avoid these problems.

Edit2 (.NET 4.5): The content of the DropDownList can be based on ToString() and not of DisplayMemberPath, while DisplayMemberPath specifies the member for the selected and displayed item only.

If the width and height are not specified WPF uses 60% of the width of you primary screen (where the windows is actually displayed) and 60% of its height. This is done by the Application singleton where all Windows ultimately register themselfes.