diff --git a/Tests/test_file_jpeg2k.py b/Tests/test_file_jpeg2k.py
index ba9896c1dff..0f045fba1ba 100644
--- a/Tests/test_file_jpeg2k.py
+++ b/Tests/test_file_jpeg2k.py
@@ -230,16 +230,14 @@ def test_header_errors() -> None:
 
 
 def test_oversized_box_length() -> None:
-    # A box declaring a 64-bit length that ends beyond the maximum seekable
-    # position must be rejected instead of raising OverflowError from seek()
+    # Do not raise an OverflowError() when seeking to the end of a box
     data = (
         b"\x00\x00\x00\x0cjP  \x0d\x0a\x87\x0a"  # JP2 signature box
-        + struct.pack(">I4s", 1, b"\x00\x00\x00\x00")  # box with 64-bit length
-        + struct.pack(">Q", 0xFFFFFFFFFF000004)  # length beyond the seek range
+        + struct.pack(">I4s", 1, b"")  # box with 64-bit length
+        + struct.pack(">Q", 16 + 2**63)  # length beyond the seek range
     )
-    with pytest.raises(UnidentifiedImageError):
-        with Image.open(BytesIO(data)):
-            pass
+    with pytest.raises(SyntaxError, match="Box length too large"):
+        Jpeg2KImagePlugin.Jpeg2KImageFile(BytesIO(data))
 
 
 def test_layers_type(card: ImageFile.ImageFile, tmp_path: Path) -> None:
diff --git a/src/PIL/Jpeg2KImagePlugin.py b/src/PIL/Jpeg2KImagePlugin.py
index d2d30567de3..e2f70d11fdd 100644
--- a/src/PIL/Jpeg2KImagePlugin.py
+++ b/src/PIL/Jpeg2KImagePlugin.py
@@ -83,6 +83,9 @@ def has_next_box(self) -> bool:
     def next_box_type(self) -> bytes:
         # Skip the rest of the box if it has not been read
         if self.remaining_in_box > 0:
+            if self.remaining_in_box >= 2**63:
+                msg = "Box length too large"
+                raise SyntaxError(msg)
             self.fp.seek(self.remaining_in_box, os.SEEK_CUR)
         self.remaining_in_box = -1
 
@@ -98,11 +101,6 @@ def next_box_type(self) -> bytes:
             msg = "Invalid header length"
             raise SyntaxError(msg)
 
-        # Reject a box whose contents end beyond the maximum seekable position
-        if self.fp.tell() + lbox - hlen >= 2**63:
-            msg = "Invalid box length"
-            raise SyntaxError(msg)
-
         self.remaining_in_box = lbox - hlen
         return tbox