diff --git a/Lib/test/test_ssl.py b/Lib/test/test_ssl.py
index 0b8cef6..8c84657 100644
--- a/Lib/test/test_ssl.py
+++ b/Lib/test/test_ssl.py
@@ -1433,12 +1433,17 @@ def test_load_verify_cadata(self):
         ctx = ssl.SSLContext(ssl.PROTOCOL_TLS_CLIENT)
         self.assertRaises(TypeError, ctx.load_verify_locations, cadata=object)
 
-        with self.assertRaisesRegex(ssl.SSLError, "no start line"):
+        with self.assertRaisesRegex(
+            ssl.SSLError,
+            "no start line: cadata does not contain a certificate"
+        ):
             ctx.load_verify_locations(cadata="broken")
-        with self.assertRaisesRegex(ssl.SSLError, "not enough data"):
+        with self.assertRaisesRegex(
+            ssl.SSLError,
+            "not enough data: cadata does not contain a certificate"
+        ):
             ctx.load_verify_locations(cadata=b"broken")
 
-
     @unittest.skipIf(Py_DEBUG_WIN32, "Avoid mixing debug/release CRT on Windows")
     def test_load_dh_params(self):
         ctx = ssl.SSLContext(ssl.PROTOCOL_TLS_SERVER)
diff --git a/Misc/NEWS.d/next/Library/2021-04-23-11-54-38.bpo-43920.cJMQ2D.rst b/Misc/NEWS.d/next/Library/2021-04-23-11-54-38.bpo-43920.cJMQ2D.rst
new file mode 100644
index 0000000..28ff0fb
--- /dev/null
+++ b/Misc/NEWS.d/next/Library/2021-04-23-11-54-38.bpo-43920.cJMQ2D.rst
@@ -0,0 +1,2 @@
+OpenSSL 3.0.0: :meth:`~ssl.SSLContext.load_verify_locations` now returns a
+consistent error message when cadata contains no valid certificate.
diff --git a/Modules/_ssl.c b/Modules/_ssl.c
index e28c128..0daa13e 100644
--- a/Modules/_ssl.c
+++ b/Modules/_ssl.c
@@ -3791,7 +3791,7 @@ _add_ca_certs(PySSLContext *self, const void *data, Py_ssize_t len,
 {
     BIO *biobuf = NULL;
     X509_STORE *store;
-    int retval = 0, err, loaded = 0;
+    int retval = -1, err, loaded = 0;
 
     assert(filetype == SSL_FILETYPE_ASN1 || filetype == SSL_FILETYPE_PEM);
 
@@ -3845,23 +3845,32 @@ _add_ca_certs(PySSLContext *self, const void *data, Py_ssize_t len,
     }
 
     err = ERR_peek_last_error();
-    if ((filetype == SSL_FILETYPE_ASN1) &&
-            (loaded > 0) &&
-            (ERR_GET_LIB(err) == ERR_LIB_ASN1) &&
-            (ERR_GET_REASON(err) == ASN1_R_HEADER_TOO_LONG)) {
+    if (loaded == 0) {
+        const char *msg = NULL;
+        if (filetype == SSL_FILETYPE_PEM) {
+            msg = "no start line: cadata does not contain a certificate";
+        } else {
+            msg = "not enough data: cadata does not contain a certificate";
+        }
+        _setSSLError(get_state_ctx(self), msg, 0, __FILE__, __LINE__);
+        retval = -1;
+    } else if ((filetype == SSL_FILETYPE_ASN1) &&
+                    (ERR_GET_LIB(err) == ERR_LIB_ASN1) &&
+                    (ERR_GET_REASON(err) == ASN1_R_HEADER_TOO_LONG)) {
         /* EOF ASN1 file, not an error */
         ERR_clear_error();
         retval = 0;
     } else if ((filetype == SSL_FILETYPE_PEM) &&
-                   (loaded > 0) &&
                    (ERR_GET_LIB(err) == ERR_LIB_PEM) &&
                    (ERR_GET_REASON(err) == PEM_R_NO_START_LINE)) {
         /* EOF PEM file, not an error */
         ERR_clear_error();
         retval = 0;
-    } else {
+    } else if (err != 0) {
         _setSSLError(get_state_ctx(self), NULL, 0, __FILE__, __LINE__);
         retval = -1;
+    } else {
+        retval = 0;
     }
 
     BIO_free(biobuf);