diff --git a/src/connect/content_stream.py b/src/connect/content_stream.py
index 1b93656..5644d58 100644
--- a/src/connect/content_stream.py
+++ b/src/connect/content_stream.py
@@ -55,7 +55,7 @@ class BoundAsyncStream(AsyncByteStream):
 
     """
 
-    _stream: AsyncIterable[bytes]
+    _stream: AsyncIterable[bytes] | None
     _closed: bool
 
     def __init__(self, stream: AsyncIterable[bytes]) -> None:
@@ -70,13 +70,19 @@ def __init__(self, stream: AsyncIterable[bytes]) -> None:
 
     async def __aiter__(self) -> AsyncIterator[bytes]:
         """Asynchronous iterator method to read byte chunks from the stream."""
+        if self._stream is None:
+            return
+
         try:
             with map_httpcore_exceptions():
                 async for chunk in self._stream:
                     yield chunk
-        except BaseException as exc:
-            await self.aclose()
-            raise exc
+        except Exception as exc:
+            try:
+                await self.aclose()
+            except Exception as close_exc:
+                raise ExceptionGroup("Multiple errors occurred", [exc, close_exc]) from exc
+            raise
 
     async def aclose(self) -> None:
         """Asynchronously close the stream."""
@@ -84,10 +90,14 @@ async def aclose(self) -> None:
             return
 
         self._closed = True
-        with map_httpcore_exceptions():
-            aclose = get_acallable_attribute(self._stream, "aclose")
-            if aclose:
-                await aclose()
+        try:
+            if self._stream is not None:
+                with map_httpcore_exceptions():
+                    aclose = get_acallable_attribute(self._stream, "aclose")
+                    if aclose:
+                        await aclose()
+        finally:
+            self._stream = None
 
 
 class AsyncDataStream[T]:
@@ -102,7 +112,7 @@ class AsyncDataStream[T]:
 
     """
 
-    _stream: AsyncIterable[T]
+    _stream: AsyncIterable[T] | None
     _aclose_func: Callable[..., Awaitable[None]] | None
     _closed: bool
 
@@ -128,12 +138,18 @@ async def __aiter__(self) -> AsyncIterator[T]:
             Propagates any exception raised during iteration after ensuring the stream is closed.
 
         """
+        if self._stream is None:
+            return
+
         try:
             async for part in self._stream:
                 yield part
-        except BaseException as exc:
-            await self.aclose()
-            raise exc
+        except Exception as exc:
+            try:
+                await self.aclose()
+            except Exception as close_exc:
+                raise ExceptionGroup("Multiple errors occurred", [exc, close_exc]) from exc
+            raise
 
     async def aclose(self) -> None:
         """Asynchronously closes the underlying stream.
@@ -149,10 +165,13 @@ async def aclose(self) -> None:
             return
 
         self._closed = True
-        if self._aclose_func:
-            await self._aclose_func()
-
-        elif hasattr(self._stream, "aclose"):
-            aclose = get_acallable_attribute(self._stream, "aclose")
-            if aclose:
-                await aclose()
+        try:
+            if self._aclose_func:
+                await self._aclose_func()
+            elif self._stream is not None:
+                aclose = get_acallable_attribute(self._stream, "aclose")
+                if aclose:
+                    await aclose()
+        finally:
+            self._stream = None
+            self._aclose_func = None