diff --git a/src/werkzeug/routing/rules.py b/src/werkzeug/routing/rules.py
index 2dad31dd3..6819218e9 100644
--- a/src/werkzeug/routing/rules.py
+++ b/src/werkzeug/routing/rules.py
@@ -623,6 +623,7 @@ def _parse_rule(self, rule: str) -> t.Iterable[RulePart]:
                 self._trace.append((False, data["static"]))
                 content += data["static"] if static else re.escape(data["static"])
 
+            haspath = False
             if data["variable"] is not None:
                 if static:
                     # Switching content to represent regex, hence the need to escape
@@ -640,6 +641,7 @@ def _parse_rule(self, rule: str) -> t.Iterable[RulePart]:
                 convertor_number += 1
                 argument_weights.append(convobj.weight)
                 self._trace.append((True, data["variable"]))
+                haspath = data["converter"] == "path"
 
             if data["slash"] is not None:
                 self._trace.append((False, "/"))
@@ -651,7 +653,7 @@ def _parse_rule(self, rule: str) -> t.Iterable[RulePart]:
                     weight = Weighting(
                         -len(static_weights),
                         static_weights,
-                        -len(argument_weights),
+                        float("+inf") if haspath else len(argument_weights),
                         argument_weights,
                     )
                     yield RulePart(
@@ -681,7 +683,7 @@ def _parse_rule(self, rule: str) -> t.Iterable[RulePart]:
         weight = Weighting(
             -len(static_weights),
             static_weights,
-            -len(argument_weights),
+            float("+inf") if haspath else len(argument_weights),
             argument_weights,
         )
         yield RulePart(
diff --git a/tests/test_routing.py b/tests/test_routing.py
index 02db898d6..2c715ebb0 100644
--- a/tests/test_routing.py
+++ b/tests/test_routing.py
@@ -381,7 +381,8 @@ def test_greedy():
         [
             r.Rule("/foo", endpoint="foo"),
             r.Rule("/<path:bar>", endpoint="bar"),
-            r.Rule("/<path:bar>/<path:blub>", endpoint="bar"),
+            r.Rule("/<path:bar>/<blub>", endpoint="bar"),
+            r.Rule("/<baz>/static", endpoint="oops"),
         ]
     )
     adapter = map.bind("example.org", "/")
@@ -389,12 +390,54 @@ def test_greedy():
     assert adapter.match("/foo") == ("foo", {})
     assert adapter.match("/blub") == ("bar", {"bar": "blub"})
     assert adapter.match("/he/he") == ("bar", {"bar": "he", "blub": "he"})
+    assert adapter.match("/he/static") == ("oops", {"baz": "he"})
+
+    assert adapter.build("foo", {}) == "/foo"
+    assert adapter.build("bar", {"bar": "blub"}) == "/blub"
+    assert adapter.build("bar", {"bar": "blub", "blub": "bar"}) == "/blub/bar"
+
+
+def test_greedy_double_paths():
+    # two back to back paths do not have any meaning and should
+    # probably cause an error in _parse_rule
+    map = r.Map(
+        [
+            r.Rule("/foo", endpoint="foo"),
+            r.Rule("/<path:bar>", endpoint="bar"),
+            r.Rule("/<path:bar>/<path:blub>", endpoint="bar"),
+        ]
+    )
+    adapter = map.bind("example.org", "/")
+
+    assert adapter.match("/foo") == ("foo", {})
+    assert adapter.match("/blub") == ("bar", {"bar": "blub"})
+    assert adapter.match("/he/he") == ("bar", {"bar": "he/he"})
+    # can't match ("bar", {"bar": "he", "blub": "he"}) without breaking static matching
+    # use a rule like "/<path:bar/<blub>" instead
 
     assert adapter.build("foo", {}) == "/foo"
     assert adapter.build("bar", {"bar": "blub"}) == "/blub"
     assert adapter.build("bar", {"bar": "blub", "blub": "bar"}) == "/blub/bar"
 
 
+def test_static_priority():
+    # see https://github.com/pallets/werkzeug/issues/2924
+    map = r.Map(
+        [
+            r.Rule("/<path:dyn2>/<dyn1>", endpoint="file"),
+            r.Rule("/<dyn1>/statn", endpoint="stat"),
+        ],
+    )
+
+    adapter = map.bind("example.org", "/")
+
+    assert adapter.match("/d2/d1", method="GET") == (
+        "file",
+        {"dyn2": "d2", "dyn1": "d1"},
+    )
+    assert adapter.match("/d1/statn", method="GET") == ("stat", {"dyn1": "d1"})
+
+
 def test_path():
     map = r.Map(
         [