TST: consistent result in dropping NA from CSV

Closes #21131

pandas/tests/io/parser/test_na_values.py

@@ -536,3 +536,31 @@ def test_cast_NA_to_bool_raises_error(all_parsers, data, na_values):
             dtype={"a": "bool"},
             na_values=na_values,
         )
+
+
+def test_str_nan_dropped(all_parsers):
+    # see gh-21131
+    parser = all_parsers
+
+    data = """File: small.csv,,
+10010010233,0123,654
+foo,,bar
+01001000155,4530,898"""
+
+    result = parser.read_csv(
+        StringIO(data),
+        header=None,
+        names=["col1", "col2", "col3"],
+        dtype={"col1": str, "col2": str, "col3": str},
+    ).dropna()
+
+    expected = DataFrame(
+        {
+            "col1": ["10010010233", "01001000155"],
+            "col2": ["0123", "4530"],
+            "col3": ["654", "898"],
+        },
+        index=[1, 3],
+    )
+
+    tm.assert_frame_equal(result, expected)