diff --git a/cpp/src/arrow/csv/parser_test.cc b/cpp/src/arrow/csv/parser_test.cc
index 960a69c59db5d..41c315165b97d 100644
--- a/cpp/src/arrow/csv/parser_test.cc
+++ b/cpp/src/arrow/csv/parser_test.cc
@@ -175,6 +175,13 @@ void AssertParsePartial(BlockParser& parser, const std::string& str,
   ASSERT_EQ(parsed_size, expected_size);
 }
 
+void AssertParsePartial(BlockParser& parser, const std::vector<std::string_view>& data,
+                        uint32_t expected_size) {
+  uint32_t parsed_size = static_cast<uint32_t>(-1);
+  ASSERT_OK(parser.Parse(data, &parsed_size));
+  ASSERT_EQ(parsed_size, expected_size);
+}
+
 void AssertLastRowEq(const BlockParser& parser,
                      const std::vector<std::string>& expected) {
   std::vector<std::string> values;
@@ -376,6 +383,19 @@ TEST(BlockParser, TruncatedData) {
   }
 }
 
+TEST(BlockParser, TruncatedDataViews) {
+  // If non-last block is truncated, parsing stops
+  BlockParser parser(ParseOptions::Defaults(), /*num_cols=*/3);
+  AssertParsePartial(parser, Views({"a,b,", "c\n"}), 0);
+  // (XXX should we guarantee this one below?)
+  AssertParsePartial(parser, Views({"a,b,c\nd,", "e,f\n"}), 6);
+
+  // More sophisticated: non-last block ends on some newline inside a quoted string
+  // (terse reproducer of gh-39857)
+  AssertParsePartial(parser, Views({"a,b,\"c\n", "\"\n"}), 0);
+  AssertParsePartial(parser, Views({"a,b,c\n\"d\n", "\",e,f\n"}), 6);
+}
+
 TEST(BlockParser, Final) {
   // Tests for ParseFinal()
   BlockParser parser(ParseOptions::Defaults());
diff --git a/cpp/src/arrow/csv/reader.cc b/cpp/src/arrow/csv/reader.cc
index 332fad054fea3..cf009f2227fce 100644
--- a/cpp/src/arrow/csv/reader.cc
+++ b/cpp/src/arrow/csv/reader.cc
@@ -263,8 +263,14 @@ class SerialBlockReader : public BlockReader {
       DCHECK_GE(nbytes, 0);
       auto offset = nbytes - bytes_before_buffer;
       if (offset < 0) {
-        // Should not happen
-        return Status::Invalid("CSV parser got out of sync with chunker");
+        // This can happen if `newlines_in_values` is not enabled and
+        // `partial + completion` ends with a newline inside a quoted string.
+        // In this case, the BlockParser stops at the truncated data in the first
+        // block (see gh-39857).
+        return Status::Invalid(
+            "CSV parser got out of sync with chunker. This can mean the data file "
+            "contains cell values spanning multiple lines; please consider enabling "
+            "the option 'newlines_in_values'.");
       }
       partial_ = SliceBuffer(buffer_, offset);
       buffer_ = next_buffer;