diff --git a/tonic/src/codec/decode.rs b/tonic/src/codec/decode.rs
index b4dda4729..66bc14e8a 100644
--- a/tonic/src/codec/decode.rs
+++ b/tonic/src/codec/decode.rs
@@ -39,6 +39,7 @@ impl<T> Unpin for Streaming<T> {}
 enum State {
     ReadHeader,
     ReadBody { compression: bool, len: usize },
+    Error,
 }
 
 #[derive(Debug)]
@@ -311,6 +312,10 @@ impl<T> Stream for Streaming<T> {
 
     fn poll_next(mut self: Pin<&mut Self>, cx: &mut Context<'_>) -> Poll<Option<Self::Item>> {
         loop {
+            if let State::Error = &self.state {
+                return Poll::Ready(None);
+            }
+
             // FIXME: implement the ability to poll trailers when we _know_ that
             // the consumer of this stream will only poll for the first message.
             // This means we skip the poll_trailers step.
@@ -321,6 +326,7 @@ impl<T> Stream for Streaming<T> {
             let chunk = match ready!(Pin::new(&mut self.body).poll_data(cx)) {
                 Some(Ok(d)) => Some(d),
                 Some(Err(e)) => {
+                    let _ = std::mem::replace(&mut self.state, State::Error);
                     let err: crate::Error = e.into();
                     debug!("decoder inner stream error: {:?}", err);
                     let status = Status::from_error(err);