bool TryingtoStart = true;

while (TryingtoSTart)
{
    try
        {
             dbc.Open();

                 departmentDataAdapter = new SQLiteDataAdapter("select * from DEPARTMENT", dbc);
                 // etc
        }

       catch (SQLiteException ex)
       {
              DialogResult r = MetroFramework.MetroMessageBox.Show(this,  ex.Message.ToString(), "Database error", MessageBoxButtons.RetryCancel);

              if (r == DialogResult.Cancel) TryingtoStart = false;
       }

       catch (Exception exc)
       {
              DialogResult r = MetroFramework.MetroMessageBox.Show(this,  exc.Message.ToString(), "Exception", MessageBoxButtons.RetryCancel);
              if (r == DialogResult.Cancel) TryingtoStart = false;
       }

       if (!TryingtoStart) Application.Exit();
}

"Operation is not valid due to the current state of the object."

System.InvalidOperationException: Operation is not valid due to the current state of the object.
   at System.Data.SQLite.SQLiteConnection.Open()

bool TryingtoStart = true;

while (TryingtoSTart)
{

    using(var dbc = new SQLiteConnection(connectionString))
    {
        try
        {
             using(var departmentDataAdapter = new SQLiteDataAdapter("select * from DEPARTMENT", dbc))
             {
                  // etc
             }

            // Note: This row will only be executed if there where no exceptions until this point in the code
            TryingtoStart = false;
        }

        catch (SQLiteException ex)
        {

            if (MetroFramework.MetroMessageBox.Show(this,  ex.Message.ToString(), "Database error", MessageBoxButtons.RetryCancel) == DialogResult.Cancel) Application.Exit();
        }
        catch (Exception exc)
        {
            if (MetroFramework.MetroMessageBox.Show(this,  exc.Message.ToString(), "Exception", MessageBoxButtons.RetryCancel) == DialogResult.Cancel) Application.Exit();
        }
    }
}