diff --git a/Lib/test/test_nntplib.py b/Lib/test/test_nntplib.py
index d7642bc66ab346..1d1750a5be22e5 100644
--- a/Lib/test/test_nntplib.py
+++ b/Lib/test/test_nntplib.py
@@ -6,6 +6,8 @@
 import functools
 import contextlib
 import os.path
+import re
+
 from test import support
 from nntplib import NNTP, GroupInfo
 import nntplib
@@ -22,6 +24,13 @@
 TIMEOUT = 30
 certfile = os.path.join(os.path.dirname(__file__), 'keycert3.pem')
 
+if ssl is not None:
+    SSLError = ssl.SSLError
+else:
+    class SSLError(Exception):
+        """Non-existent exception class when we lack SSL support."""
+        reason = "This will never be raised."
+
 # TODO:
 # - test the `file` arg to more commands
 # - test error conditions
@@ -262,14 +271,21 @@ def is_connected():
                 return False
             return True
 
-        with self.NNTP_CLASS(self.NNTP_HOST, timeout=TIMEOUT, usenetrc=False) as server:
-            self.assertTrue(is_connected())
-            self.assertTrue(server.help())
-        self.assertFalse(is_connected())
-
-        with self.NNTP_CLASS(self.NNTP_HOST, timeout=TIMEOUT, usenetrc=False) as server:
-            server.quit()
-        self.assertFalse(is_connected())
+        try:
+            with self.NNTP_CLASS(self.NNTP_HOST, timeout=TIMEOUT, usenetrc=False) as server:
+                self.assertTrue(is_connected())
+                self.assertTrue(server.help())
+            self.assertFalse(is_connected())
+
+            with self.NNTP_CLASS(self.NNTP_HOST, timeout=TIMEOUT, usenetrc=False) as server:
+                server.quit()
+            self.assertFalse(is_connected())
+        except SSLError as ssl_err:
+            # matches "[SSL: DH_KEY_TOO_SMALL] dh key too small"
+            if re.search(r'(?i)KEY.TOO.SMALL', ssl_err.reason):
+                raise unittest.SkipTest(f"Got {ssl_err} connecting "
+                                        f"to {self.NNTP_HOST!r}")
+            raise
 
 
 NetworkedNNTPTestsMixin.wrap_methods()
@@ -290,6 +306,12 @@ def setUpClass(cls):
             try:
                 cls.server = cls.NNTP_CLASS(cls.NNTP_HOST, timeout=TIMEOUT,
                                             usenetrc=False)
+            except SSLError as ssl_err:
+                # matches "[SSL: DH_KEY_TOO_SMALL] dh key too small"
+                if re.search(r'(?i)KEY.TOO.SMALL', ssl_err.reason):
+                    raise unittest.SkipTest(f"{cls} got {ssl_err} connecting "
+                                            f"to {cls.NNTP_HOST!r}")
+                raise
             except EOFError:
                 raise unittest.SkipTest(f"{cls} got EOF error on connecting "
                                         f"to {cls.NNTP_HOST!r}")
diff --git a/Lib/test/test_ssl.py b/Lib/test/test_ssl.py
index 0aeabc10f2a97f..e47edd22401d87 100644
--- a/Lib/test/test_ssl.py
+++ b/Lib/test/test_ssl.py
@@ -17,6 +17,7 @@
 import asyncore
 import weakref
 import platform
+import re
 import functools
 try:
     import ctypes
@@ -145,6 +146,38 @@ def f(*args, **kwargs):
     else:
         return func
 
+def skip_if_openssl_cnf_minprotocol_gt_tls1(func):
+    """Skip a test if the OpenSSL config MinProtocol is > TLSv1.
+
+    OS distros with an /etc/ssl/openssl.cnf and MinProtocol set often do so to
+    require TLSv1.2 or higher (Debian Buster).  Some of our tests for older
+    protocol versions will fail under such a config.
+
+    Alternative workaround: Run this test in a process with
+    OPENSSL_CONF=/dev/null in the environment.
+    """
+    @functools.wraps(func)
+    def f(*args, **kwargs):
+        openssl_cnf = os.environ.get("OPENSSL_CONF", "/etc/ssl/openssl.cnf")
+        try:
+            with open(openssl_cnf, "r") as config:
+                for line in config:
+                    match = re.match(r"MinProtocol\s*=\s*(TLSv\d+\S*)", line)
+                    if match:
+                        tls_ver = match.group(1)
+                        if tls_ver > "TLSv1":
+                            raise unittest.SkipTest(
+                                "%s has MinProtocol = %s which is > TLSv1." %
+                                (openssl_cnf, tls_ver))
+        except (EnvironmentError, UnicodeDecodeError) as err:
+            # no config file found, etc.
+            if support.verbose:
+                sys.stdout.write("\n Could not scan %s for MinProtocol: %s\n"
+                                 % (openssl_cnf, err))
+        return func(*args, **kwargs)
+    return f
+
+
 needs_sni = unittest.skipUnless(ssl.HAS_SNI, "SNI support needed for this test")
 
 
@@ -2614,6 +2647,7 @@ def test_protocol_sslv2(self):
                                client_options=ssl.OP_NO_TLSv1)
 
         @skip_if_broken_ubuntu_ssl
+        @skip_if_openssl_cnf_minprotocol_gt_tls1
         def test_protocol_sslv23(self):
             """Connecting to an SSLv23 server with various client options"""
             if support.verbose:
@@ -2691,6 +2725,7 @@ def test_protocol_tlsv1(self):
         @skip_if_broken_ubuntu_ssl
         @unittest.skipUnless(hasattr(ssl, "PROTOCOL_TLSv1_1"),
                              "TLS version 1.1 not supported.")
+        @skip_if_openssl_cnf_minprotocol_gt_tls1
         def test_protocol_tlsv1_1(self):
             """Connecting to a TLSv1.1 server with various client options.
                Testing against older TLS versions."""
diff --git a/Misc/NEWS.d/next/Tests/2019-05-06-18-29-54.bpo-35925.gwQPuC.rst b/Misc/NEWS.d/next/Tests/2019-05-06-18-29-54.bpo-35925.gwQPuC.rst
new file mode 100644
index 00000000000000..428326cdfe5bba
--- /dev/null
+++ b/Misc/NEWS.d/next/Tests/2019-05-06-18-29-54.bpo-35925.gwQPuC.rst
@@ -0,0 +1 @@
+Skip specific nntplib and ssl networking tests when they would otherwise fail due to a modern OS or distro with a default OpenSSL policy of rejecting connections to servers with weak certificates or disabling TLS below TLSv1.2.