For "normal" key-click tests (like entering text in a line-edit), it is not necessary to show the window. This is in line with what you'd expect if you sent key events to a hidden widget during normal running of the application.
But for testing shortcuts, the target window must be shown - which is again in line with what you'd expect. A keyboard shortcut should not activate commands during normal running if the target window is not visible.
So your setup code should probably include something like this:
    self.window.show()
    QTest.qWaitForWindowShown(self.window)
The qWaitForWindowShown call is necessary on systems where windows are shown asynchronously (for Qt5, use qWaitForWindowExposed).
EDIT:
Here's a test script that works for me:
import unittest
from PySide.QtCore import Qt
from PySide.QtGui import QApplication, QMainWindow, QLineEdit
from PySide.QtTest import QTest
class Window(QMainWindow):
    def __init__(self):
        QMainWindow.__init__(self)
        menu = self.menuBar().addMenu('File')
        menu.addAction('Test', self.handleTest, 'Ctrl+N')
        self.edit = QLineEdit(self)
        self.setCentralWidget(self.edit)
    def handleTest(self):
        self.edit.setText('test')
class AppTestCase(unittest.TestCase):
    def setUp(self):
        qApp = QApplication.instance()
        if qApp is None:
            self.app = QApplication([])
        else:
            self.app = qApp
class WindowTestCase(AppTestCase):
    def setUp(self):
        super(WindowTestCase, self).setUp()
        self.window = Window()
        self.window.show()
        QTest.qWaitForWindowShown(self.window)
    def test_input_object_in_new_file(self):
        text = 'test'
        self.assertNotEqual(text, self.window.edit.text())
        QTest.keyClicks(self.window, 'n', Qt.ControlModifier)
        self.assertEqual(text, self.window.edit.text())
    def test_enter_text(self):
        text = 'foobar'
        self.assertNotEqual(text, self.window.edit.text())
        QTest.keyClicks(self.window.edit, text)
        self.assertEqual(text, self.window.edit.text())
if __name__ == "__main__":
    unittest.main()
UPDATE:
Here's a PyQt5 version of the above script:
import unittest
from PyQt5.QtCore import Qt
from PyQt5.QtWidgets import QApplication, QMainWindow, QLineEdit
from PyQt5.QtTest import QTest
class Window(QMainWindow):
    def __init__(self):
        QMainWindow.__init__(self)
        menu = self.menuBar().addMenu('File')
        menu.addAction('Test', self.handleTest, 'Ctrl+N')
        self.edit = QLineEdit(self)
        self.setCentralWidget(self.edit)
    def handleTest(self):
        self.edit.setText('test')
class AppTestCase(unittest.TestCase):
    def setUp(self):
        qApp = QApplication.instance()
        if qApp is None:
            self.app = QApplication([''])
        else:
            self.app = qApp
class WindowTestCase(AppTestCase):
    def setUp(self):
        super(WindowTestCase, self).setUp()
        self.window = Window()
        self.window.show()
        QTest.qWaitForWindowExposed(self.window)
    def test_input_object_in_new_file(self):
        text = 'test'
        self.assertNotEqual(text, self.window.edit.text())
        QTest.keyClicks(self.window, 'n', Qt.ControlModifier)
        self.assertEqual(text, self.window.edit.text())
    def test_enter_text(self):
        text = 'foobar'
        self.assertNotEqual(text, self.window.edit.text())
        QTest.keyClicks(self.window.edit, text)
        self.assertEqual(text, self.window.edit.text())
if __name__ == "__main__":
    unittest.main()