diff --git a/news/6158.bugfix b/news/6158.bugfix
new file mode 100644
index 00000000000..fe5a0352f9b
--- /dev/null
+++ b/news/6158.bugfix
@@ -0,0 +1 @@
+Fix a crash when using --no-cache-dir with PEP 517 distributions
diff --git a/news/6171.bugfix b/news/6171.bugfix
new file mode 100644
index 00000000000..fe5a0352f9b
--- /dev/null
+++ b/news/6171.bugfix
@@ -0,0 +1 @@
+Fix a crash when using --no-cache-dir with PEP 517 distributions
diff --git a/src/pip/_internal/wheel.py b/src/pip/_internal/wheel.py
index 93b4768622a..3f200b77edc 100644
--- a/src/pip/_internal/wheel.py
+++ b/src/pip/_internal/wheel.py
@@ -840,12 +840,6 @@ def build(
             newly built wheel, in preparation for installation.
         :return: True if all the wheels built correctly.
         """
-        # TODO: This check fails if --no-cache-dir is set. And yet we
-        #       might be able to build into the ephemeral cache, surely?
-        building_is_possible = self._wheel_dir or (
-            autobuilding and self.wheel_cache.cache_dir
-        )
-        assert building_is_possible
 
         buildset = []
         format_control = self.finder.format_control
@@ -884,6 +878,13 @@ def build(
         if not buildset:
             return []
 
+        # Is any wheel build not using the ephemeral cache?
+        if any(not ephem_cache for _, ephem_cache in buildset):
+            have_directory_for_build = self._wheel_dir or (
+                autobuilding and self.wheel_cache.cache_dir
+            )
+            assert have_directory_for_build
+
         # TODO by @pradyunsg
         # Should break up this method into 2 separate methods.
 
diff --git a/tests/functional/test_install_wheel.py b/tests/functional/test_install_wheel.py
index 9937d5a0af3..e09fdce850a 100644
--- a/tests/functional/test_install_wheel.py
+++ b/tests/functional/test_install_wheel.py
@@ -420,3 +420,11 @@ def test_wheel_compile_syntax_error(script, data):
     result = script.pip('install', '--compile', package, '--no-index')
     assert 'yield from' not in result.stdout
     assert 'SyntaxError: ' not in result.stdout
+
+
+def test_wheel_install_with_no_cache_dir(script, tmpdir, data):
+    """Check wheel installations work, even with no cache.
+    """
+    package = data.packages.join("simple.dist-0.1-py2.py3-none-any.whl")
+    result = script.pip('install', '--no-cache-dir', '--no-index', package)
+    result.assert_installed('simpledist', editable=False)
diff --git a/tests/functional/test_pep517.py b/tests/functional/test_pep517.py
index 52e3acded90..a1a45d27bb9 100644
--- a/tests/functional/test_pep517.py
+++ b/tests/functional/test_pep517.py
@@ -109,3 +109,17 @@ def test_pep517_backend_requirements_already_satisfied(script, tmpdir, data):
         project_dir,
     )
     assert 'Installing backend dependencies:' not in result.stdout
+
+
+def test_pep517_install_with_no_cache_dir(script, tmpdir, data):
+    """Check builds with a custom backends work, even with no cache.
+    """
+    project_dir = make_project(
+        tmpdir, requires=['test_backend'],
+        backend="test_backend"
+    )
+    result = script.pip(
+        'install', '--no-cache-dir', '--no-index', '-f', data.backends,
+        project_dir,
+    )
+    result.assert_installed('project', editable=False)