diff --git a/mypy/modulefinder.py b/mypy/modulefinder.py
index ad856aef2714..40f2db91adf1 100644
--- a/mypy/modulefinder.py
+++ b/mypy/modulefinder.py
@@ -198,9 +198,7 @@ def find_module(self, id: str, *, fast_path: bool = False) -> ModuleSearchResult
             top_level = id.partition('.')[0]
             use_typeshed = True
             if top_level in self.stdlib_py_versions:
-                min_version = self.stdlib_py_versions[top_level]
-                use_typeshed = (self.options is None
-                                or typeshed_py_version(self.options) >= min_version)
+                use_typeshed = self._typeshed_has_version(top_level)
             self.results[id] = self._find_module(id, use_typeshed)
             if (not fast_path
                     and self.results[id] is ModuleNotFoundReason.NOT_FOUND
@@ -208,6 +206,13 @@ def find_module(self, id: str, *, fast_path: bool = False) -> ModuleSearchResult
                 self.results[id] = ModuleNotFoundReason.WRONG_WORKING_DIRECTORY
         return self.results[id]
 
+    def _typeshed_has_version(self, module: str) -> bool:
+        if not self.options:
+            return True
+        version = typeshed_py_version(self.options)
+        min_version, max_version = self.stdlib_py_versions[module]
+        return version >= min_version and (max_version is None or version <= max_version)
+
     def _find_module_non_stub_helper(self, components: List[str],
                                      pkg_dir: str) -> Union[OnePackageDir, ModuleNotFoundReason]:
         plausible_match = False
@@ -735,10 +740,14 @@ def compute_search_paths(sources: List[BuildSource],
                        typeshed_path=tuple(lib_path))
 
 
-def load_stdlib_py_versions(custom_typeshed_dir: Optional[str]) -> Dict[str, Tuple[int, int]]:
-    """Return dict with minimum Python versions of stdlib modules.
+def load_stdlib_py_versions(custom_typeshed_dir: Optional[str]
+                            ) -> Dict[str, Tuple[Tuple[int, int], Optional[Tuple[int, int]]]]:
+    """Return dict with minimum and maximum Python versions of stdlib modules.
+
+    The contents look like
+    {..., 'secrets': ((3, 6), None), 'symbol': (2, 7), (3, 9)), ...}
 
-    The contents look like {..., 'secrets': 3.6, 're': 2.7, ...}.
+    None means there is no maximum version.
     """
     typeshed_dir = custom_typeshed_dir or os.path.join(os.path.dirname(__file__), "typeshed")
     stdlib_dir = os.path.join(typeshed_dir, "stdlib")
@@ -751,20 +760,29 @@ def load_stdlib_py_versions(custom_typeshed_dir: Optional[str]) -> Dict[str, Tup
             line = line.strip()
             if line.startswith("#") or line == "":
                 continue
-            module, version = line.split(":")
-            major, minor = version.strip().split(".")
-            result[module] = int(major), int(minor)
+            module, version_range = line.split(":")
+            versions = version_range.split("-")
+            min_version = parse_version(versions[0])
+            max_version = (parse_version(versions[1])
+                           if len(versions) >= 2 and versions[1].strip() else None)
+            result[module] = min_version, max_version
 
     # Modules that are Python 2 only or have separate Python 2 stubs
     # have stubs in @python2/ and may need an override.
     python2_dir = os.path.join(stdlib_dir, PYTHON2_STUB_DIR)
     for fnam in os.listdir(python2_dir):
         fnam = fnam.replace(".pyi", "")
-        result[fnam] = (2, 7)
+        max_version = result.get(fnam, ((2, 7), None))[1]
+        result[fnam] = (2, 7), max_version
 
     return result
 
 
+def parse_version(version: str) -> Tuple[int, int]:
+    major, minor = version.strip().split(".")
+    return int(major), int(minor)
+
+
 def typeshed_py_version(options: Options) -> Tuple[int, int]:
     """Return Python version used for checking whether module supports typeshed."""
     # Typeshed no longer covers Python 3.x versions before 3.6, so 3.6 is
diff --git a/mypy/stubtest.py b/mypy/stubtest.py
index db8666c94e3e..7915e14f1551 100644
--- a/mypy/stubtest.py
+++ b/mypy/stubtest.py
@@ -1074,8 +1074,9 @@ def get_typeshed_stdlib_modules(custom_typeshed_dir: Optional[str]) -> List[str]
         version_info = (3, 6)
     else:
         version_info = sys.version_info[0:2]
-    for module, minver in stdlib_py_versions.items():
-        if version_info >= minver:
+    for module, versions in stdlib_py_versions.items():
+        minver, maxver = versions
+        if version_info >= minver and (maxver is None or version_info <= maxver):
             packages.add(module)
 
     if custom_typeshed_dir: