diff --git a/src/openai/_streaming.py b/src/openai/_streaming.py
index cee737f4f5..095746630b 100644
--- a/src/openai/_streaming.py
+++ b/src/openai/_streaming.py
@@ -47,8 +47,9 @@ def __stream__(self) -> Iterator[ResponseT]:
         cast_to = self._cast_to
         response = self.response
         process_data = self._client._process_response_data
+        iterator = self._iter_events()
 
-        for sse in self._iter_events():
+        for sse in iterator:
             if sse.data.startswith("[DONE]"):
                 break
 
@@ -63,6 +64,10 @@ def __stream__(self) -> Iterator[ResponseT]:
 
                 yield process_data(data=data, cast_to=cast_to, response=response)
 
+        # Ensure the entire stream is consumed
+        for sse in iterator:
+            ...
+
 
 class AsyncStream(Generic[ResponseT]):
     """Provides the core interface to iterate over an asynchronous stream response."""
@@ -97,8 +102,9 @@ async def __stream__(self) -> AsyncIterator[ResponseT]:
         cast_to = self._cast_to
         response = self.response
         process_data = self._client._process_response_data
+        iterator = self._iter_events()
 
-        async for sse in self._iter_events():
+        async for sse in iterator:
             if sse.data.startswith("[DONE]"):
                 break
 
@@ -113,6 +119,10 @@ async def __stream__(self) -> AsyncIterator[ResponseT]:
 
                 yield process_data(data=data, cast_to=cast_to, response=response)
 
+        # Ensure the entire stream is consumed
+        async for sse in iterator:
+            ...
+
 
 class ServerSentEvent:
     def __init__(