diff --git a/openmc/mesh.py b/openmc/mesh.py
index 157e782633..dda7257f45 100644
--- a/openmc/mesh.py
+++ b/openmc/mesh.py
@@ -433,6 +433,9 @@ class RegularMesh(StructuredMesh):
         cv.check_length('mesh lower_left', lower_left, 1, 3)
         self._lower_left = lower_left
 
+        if self.upper_right is not None and any(np.isclose(self.upper_right, lower_left)):
+            raise ValueError("Mesh cannot have zero thickness in any dimension")
+
     @upper_right.setter
     def upper_right(self, upper_right):
         cv.check_type('mesh upper_right', upper_right, Iterable, Real)
@@ -442,6 +445,9 @@ class RegularMesh(StructuredMesh):
         if self._width is not None:
             self._width = None
             warnings.warn("Unsetting width attribute.")
+        
+        if self.lower_left is not None and any(np.isclose(self.lower_left, upper_right)):
+            raise ValueError("Mesh cannot have zero thickness in any dimension")
 
     @width.setter
     def width(self, width):
diff --git a/tests/unit_tests/test_mesh.py b/tests/unit_tests/test_mesh.py
new file mode 100644
index 0000000000..0f9b96b0b3
--- /dev/null
+++ b/tests/unit_tests/test_mesh.py
@@ -0,0 +1,35 @@
+import openmc
+import pytest
+import numpy as np
+
+@pytest.mark.parametrize("val_left,val_right", [(0, 0), (-1., -1.), (2.0, 2)])
+def test_raises_error_when_flat(val_left, val_right):
+    """Checks that an error is raised when a mesh is flat"""
+    mesh = openmc.RegularMesh()
+
+    # Same X
+    with pytest.raises(ValueError):
+        mesh.lower_left = [val_left, -25, -25]
+        mesh.upper_right = [val_right, 25, 25]
+
+    with pytest.raises(ValueError):
+        mesh.upper_right = [val_right, 25, 25]
+        mesh.lower_left = [val_left, -25, -25]
+
+    # Same Y
+    with pytest.raises(ValueError):
+        mesh.lower_left = [-25, val_left, -25]
+        mesh.upper_right = [25, val_right, 25]
+
+    with pytest.raises(ValueError):
+        mesh.upper_right = [25, val_right, 25]
+        mesh.lower_left = [-25, val_left, -25]
+
+    # Same Z
+    with pytest.raises(ValueError):
+        mesh.lower_left = [-25, -25, val_left]
+        mesh.upper_right = [25, 25, val_right]
+
+    with pytest.raises(ValueError):
+        mesh.upper_right = [25, 25, val_right]
+        mesh.lower_left = [-25, -25, val_left]