@@ -574,6 +574,9 @@ Bug Fixes
 
   - Use more sensible fix values for invalid NAXISj header values. [#5935]
 
+  - Close file on error to avoid creating a ``ResourceWarning`` warning
+    about an unclosed file. [#6168, #6177]
+
 - ``astropy.io.misc``
 
 - ``astropy.io.registry``
 
@@ -95,6 +95,7 @@ def __init__(self, fileobj=None, mode=None, memmap=None, overwrite=False,
             self.readonly = False
             self.writeonly = False
             self.simulateonly = True
+            self.close_on_error = False
             return
         else:
             self.simulateonly = False
@@ -132,6 +133,10 @@ def __init__(self, fileobj=None, mode=None, memmap=None, overwrite=False,
         # Underlying fileobj is a file-like object, but an actual file object
         self.file_like = False
 
+        # Should the object be closed on error: see
+        # https://github.com/astropy/astropy/issues/6168
+        self.close_on_error = False
+
         # More defaults to be adjusted below as necessary
         self.compression = None
         self.readonly = False
@@ -358,6 +363,7 @@ def close(self):
         self._mmap = None
 
         self.closed = True
+        self.close_on_error = False
 
     def _maybe_close_mmap(self, refcount_delta=0):
         """
@@ -497,6 +503,7 @@ def _open_filename(self, filename, mode, overwrite):
             self._file = bz2.BZ2File(self.name, bzip2_mode)
         else:
             self._file = fileobj_open(self.name, IO_FITS_MODES[mode])
+            self.close_on_error = True
 
         # Make certain we're back at the beginning of the file
         # BZ2File does not support seek when the file is open for writing, but
 
@@ -1019,9 +1019,14 @@ def _readfrom(cls, fileobj=None, data=None, mode=None,
 
         # Make sure at least the PRIMARY HDU can be read
         read_one = hdulist._read_next_hdu()
+
         # If we're trying to read only and no header units were found,
-        # raise and exception
+        # raise an exception
         if not read_one and mode in ('readonly', 'denywrite'):
+            # Close the file if necessary (issue #6168)
+            if hdulist._file.close_on_error:
+                hdulist._file.close()
+
             raise IOError('Empty or corrupt FITS file')
 
         if not lazy_load_hdus:
@@ -1093,6 +1098,12 @@ def _read_next_hdu(self):
                         self._read_all = True
                         return False
                     except IOError:
+                        # Close the file: see
+                        # https://github.com/astropy/astropy/issues/6168
+                        #
+                        if self._file.close_on_error:
+                            self._file.close()
+
                         if fileobj.writeonly:
                             self._read_all = True
                             return False
 
@@ -11,6 +11,7 @@
 
 from ..verify import VerifyError
 from ....extern.six.moves import range
+from ....extern import six
 from ....io import fits
 from ....tests.helper import raises, catch_warnings, ignore_warnings
 from ....utils.exceptions import AstropyUserWarning, AstropyDeprecationWarning
@@ -913,3 +914,53 @@ def test_proper_error_raised_on_non_fits_file_with_unicode(self):
         # This should raise an IOError because there is no end card.
         with pytest.raises(IOError):
             fits.open(filename)
+
+    @pytest.mark.skipif(six.PY2,
+                        reason='ResourceWarning is not created in Python 2')
+    def test_no_resource_warning_raised_on_non_fits_file(self):
+        """
+        Regression test for https://github.com/astropy/astropy/issues/6168
+
+        The ResourceWarning shows up when (in python 3+) you try to
+        open a non-FITS file when using a filename.
+        """
+
+        # To avoid creating the file multiple times the tests are
+        # all included in one test file. See the discussion to the
+        # PR at https://github.com/astropy/astropy/issues/6168
+        #
+        filename = self.temp('not-fits.fits')
+        with open(filename, mode='w') as f:
+            f.write('# header line\n')
+            f.write('0.1 0.2\n')
+
+        # Opening the file should raise an OSError however the file
+        # is opened (there are two distinct code paths, depending on
+        # whether ignore_missing_end is True or False).
+        #
+        # Explicit tests are added to make sure the file handle is not
+        # closed when passed in to fits.open. In this case the ResourceWarning
+        # was not raised, but a check is still included.
+        #
+        with catch_warnings(ResourceWarning) as ws:
+
+            # Make sure that files opened by the user are not closed
+            with open(filename, mode='rb') as f:
+                with pytest.raises(OSError):
+                    fits.open(f, ignore_missing_end=False)
+
+                assert not f.closed
+
+            with open(filename, mode='rb') as f:
+                with pytest.raises(OSError):
+                    fits.open(f, ignore_missing_end=True)
+
+                assert not f.closed
+
+            with pytest.raises(OSError):
+                fits.open(filename, ignore_missing_end=False)
+
+            with pytest.raises(OSError):
+                fits.open(filename, ignore_missing_end=True)
+
+        assert len(ws) == 0