diff --git a/Lib/test/test_unittest/test_discovery.py b/Lib/test/test_unittest/test_discovery.py
index 004898ed431834..dcb72d73efceab 100644
--- a/Lib/test/test_unittest/test_discovery.py
+++ b/Lib/test/test_unittest/test_discovery.py
@@ -571,7 +571,7 @@ def _get_module_from_name(name):
         result = unittest.TestResult()
         suite.run(result)
         self.assertEqual(len(result.skipped), 1)
-        self.assertEqual(result.testsRun, 1)
+        self.assertEqual(result.testsRun, 0)
         self.assertEqual(import_calls, ['my_package'])
 
         # Check picklability
diff --git a/Lib/test/test_unittest/test_skipping.py b/Lib/test/test_unittest/test_skipping.py
index f146dcac18ecc0..1a6af06d32b433 100644
--- a/Lib/test/test_unittest/test_skipping.py
+++ b/Lib/test/test_unittest/test_skipping.py
@@ -103,16 +103,16 @@ def test_dont_skip(self): pass
             result = LoggingResult(events)
             self.assertIs(suite.run(result), result)
             self.assertEqual(len(result.skipped), 1)
-            expected = ['startTest', 'addSkip', 'stopTest',
-                        'startTest', 'addSuccess', 'stopTest']
+            expected = ['addSkip', 'stopTest', 'startTest',
+                        'addSuccess', 'stopTest']
             self.assertEqual(events, expected)
-            self.assertEqual(result.testsRun, 2)
+            self.assertEqual(result.testsRun, 1)
             self.assertEqual(result.skipped, [(test_do_skip, "testing")])
             self.assertTrue(result.wasSuccessful())
 
             events = []
             result = test_do_skip.run()
-            self.assertEqual(events, ['startTestRun', 'startTest', 'addSkip',
+            self.assertEqual(events, ['startTestRun', 'addSkip',
                                       'stopTest', 'stopTestRun'])
             self.assertEqual(result.skipped, [(test_do_skip, "testing")])
 
@@ -135,13 +135,13 @@ def test_1(self):
         test = Foo("test_1")
         suite = unittest.TestSuite([test])
         self.assertIs(suite.run(result), result)
-        self.assertEqual(events, ['startTest', 'addSkip', 'stopTest'])
+        self.assertEqual(events, ['addSkip', 'stopTest'])
         self.assertEqual(result.skipped, [(test, "testing")])
         self.assertEqual(record, [])
 
         events = []
         result = test.run()
-        self.assertEqual(events, ['startTestRun', 'startTest', 'addSkip',
+        self.assertEqual(events, ['startTestRun', 'addSkip',
                                   'stopTest', 'stopTestRun'])
         self.assertEqual(result.skipped, [(test, "testing")])
         self.assertEqual(record, [])
diff --git a/Lib/unittest/case.py b/Lib/unittest/case.py
index 001b640dc43ad6..811557498bb30e 100644
--- a/Lib/unittest/case.py
+++ b/Lib/unittest/case.py
@@ -606,7 +606,6 @@ def run(self, result=None):
         else:
             stopTestRun = None
 
-        result.startTest(self)
         try:
             testMethod = getattr(self, self._testMethodName)
             if (getattr(self.__class__, "__unittest_skip__", False) or
@@ -617,6 +616,9 @@ def run(self, result=None):
                 _addSkip(result, self, skip_why)
                 return result
 
+            # Increase the number of tests only if it hasn't been skipped
+            result.startTest(self)
+
             expecting_failure = (
                 getattr(self, "__unittest_expecting_failure__", False) or
                 getattr(testMethod, "__unittest_expecting_failure__", False)
diff --git a/Lib/unittest/result.py b/Lib/unittest/result.py
index 3ace0a5b7bf2ef..9e56f658027f4d 100644
--- a/Lib/unittest/result.py
+++ b/Lib/unittest/result.py
@@ -97,10 +97,12 @@ def _restoreStdout(self):
 
             sys.stdout = self._original_stdout
             sys.stderr = self._original_stderr
-            self._stdout_buffer.seek(0)
-            self._stdout_buffer.truncate()
-            self._stderr_buffer.seek(0)
-            self._stderr_buffer.truncate()
+            if self._stdout_buffer is not None:
+                self._stdout_buffer.seek(0)
+                self._stdout_buffer.truncate()
+            if self._stderr_buffer is not None:
+                self._stderr_buffer.seek(0)
+                self._stderr_buffer.truncate()
 
     def stopTestRun(self):
         """Called once after all tests are executed.
diff --git a/Misc/NEWS.d/next/Library/2023-07-11-08-56-40.gh-issue-106584.g-SBtC.rst b/Misc/NEWS.d/next/Library/2023-07-11-08-56-40.gh-issue-106584.g-SBtC.rst
new file mode 100644
index 00000000000000..a13b61bf1c121b
--- /dev/null
+++ b/Misc/NEWS.d/next/Library/2023-07-11-08-56-40.gh-issue-106584.g-SBtC.rst
@@ -0,0 +1,2 @@
+Fix exit code for ``unittest`` if all tests are skipped.
+Patch by Egor Eliseev.