diff --git a/Tests/test_file_jpeg2k.py b/Tests/test_file_jpeg2k.py
index 5cd4f14c45e..6b58ec9a8a2 100644
--- a/Tests/test_file_jpeg2k.py
+++ b/Tests/test_file_jpeg2k.py
@@ -229,23 +229,14 @@ def test_header_errors() -> None:
             pass
 
 
-def test_oversized_box_length() -> None:
-    # Do not raise an OverflowError() when seeking to the end of a box
+@pytest.mark.parametrize("tbox", (b"", b"jp2h"))
+def test_oversized_box_length(tbox: bytes) -> None:
+    # Do not raise an OverflowError() when seeking to the end of a box,
+    # or when reading the contents of a box
     data = (
         b"\x00\x00\x00\x0cjP  \x0d\x0a\x87\x0a"  # JP2 signature box
-        + struct.pack(">I4s", 1, b"")  # box with 64-bit length
-        + struct.pack(">Q", 2**63 - 12)  # length beyond the seek range
-    )
-    with pytest.raises(SyntaxError, match="Box length too large"):
-        Jpeg2KImagePlugin.Jpeg2KImageFile(BytesIO(data))
-
-
-def test_oversized_box_length_when_reading() -> None:
-    # Do not raise an OverflowError() when reading the contents of a box
-    data = (
-        b"\x00\x00\x00\x0cjP  \x0d\x0a\x87\x0a"  # JP2 signature box
-        + struct.pack(">I4s", 1, b"jp2h")  # box read into with a 64-bit length
-        + struct.pack(">Q", 16 + 2**63)  # length whose contents overflow fp.read()
+        + struct.pack(">I4s", 1, tbox)  # box with 64-bit length
+        + struct.pack(">Q", 2**63 - 12)  # oversized length
     )
     with pytest.raises(SyntaxError, match="Box length too large"):
         Jpeg2KImagePlugin.Jpeg2KImageFile(BytesIO(data))
diff --git a/src/PIL/Jpeg2KImagePlugin.py b/src/PIL/Jpeg2KImagePlugin.py
index dbacb72f350..7d27fcc8fff 100644
--- a/src/PIL/Jpeg2KImagePlugin.py
+++ b/src/PIL/Jpeg2KImagePlugin.py
@@ -54,6 +54,9 @@ def _read_bytes(self, num_bytes: int) -> bytes:
         if not self._can_read(num_bytes):
             msg = "Not enough data in header"
             raise SyntaxError(msg)
+        if self.fp.tell() + num_bytes >= 2**63:
+            msg = "Box length too large"
+            raise SyntaxError(msg)
 
         data = self.fp.read(num_bytes)
         if len(data) < num_bytes:
@@ -83,6 +86,9 @@ 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.fp.tell() + 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
 
@@ -98,12 +104,6 @@ def next_box_type(self) -> bytes:
             msg = "Invalid header length"
             raise SyntaxError(msg)
 
-        # Reject a box whose contents would extend beyond the maximum position
-        # that can be sought or read, before it is stored and later consumed
-        if self.fp.tell() + lbox - hlen >= 2**63:
-            msg = "Box length too large"
-            raise SyntaxError(msg)
-
         self.remaining_in_box = lbox - hlen
         return tbox