diff --git a/doc/source/whatsnew/v0.24.0.txt b/doc/source/whatsnew/v0.24.0.txt
index 3a8c81b5e9281..100d2a406c577 100644
--- a/doc/source/whatsnew/v0.24.0.txt
+++ b/doc/source/whatsnew/v0.24.0.txt
@@ -243,6 +243,10 @@ Documentation Changes
 Bug Fixes
 ~~~~~~~~~
 
+**Data-type specific**
+
+- Bug in :meth:`Series.interpolate()` where appropiate error was not raised when interpolating with a non-numeric index column (:issue:`21662`)
+
 Categorical
 ^^^^^^^^^^^
 
diff --git a/pandas/core/generic.py b/pandas/core/generic.py
index 26c23b84a9c04..80fffd259c6ab 100644
--- a/pandas/core/generic.py
+++ b/pandas/core/generic.py
@@ -6024,6 +6024,13 @@ def interpolate(self, method='linear', axis=0, limit=None, inplace=False,
             raise NotImplementedError("Interpolation with NaNs in the index "
                                       "has not been implemented. Try filling "
                                       "those NaNs before interpolating.")
+
+        if not is_numeric_dtype(index):
+            raise ValueError("Index column must be numeric when using any "
+                            "interpolation method other than linear. Try "
+                            "setting a numeric index column before "
+                            "interpolating.")
+
         data = _maybe_transposed_self._data
         new_data = data.interpolate(method=method, axis=ax, index=index,
                                     values=_maybe_transposed_self, limit=limit,
diff --git a/pandas/tests/generic/test_generic.py b/pandas/tests/generic/test_generic.py
index 533bff0384ad9..a41b680f8648a 100644
--- a/pandas/tests/generic/test_generic.py
+++ b/pandas/tests/generic/test_generic.py
@@ -1008,3 +1008,20 @@ def test_pipe_panel(self):
 
             with pytest.raises(ValueError):
                 result = wp.pipe((f, 'y'), x=1, y=1)
+
+    def test_interpolate(self):
+        # GH 21662
+        # Using a numeric index column
+        df = DataFrame([0, 1, np.nan, 3], index=[1, 2, 3, 4])
+        series = Series(df[0])
+        assert series.interpolate(
+            method="quadratic").equals(series.interpolate(method="linear"))
+
+        # Using a non-numeric index column
+        df = DataFrame([0, 1, np.nan, 3], index=["A", "B", "C", "D"])
+        series = Series(df[0])
+        with pytest.raises(ValueError, match="Index column must be numeric "
+            "when using any interpolation method other than linear. Try "
+            "setting a numeric index column before interpolating."):
+
+            result = series.interpolate(method="quadratic")