diff --git a/lib/http.c b/lib/http.c
index 4344b9dae58b30..e74aba32286d0e 100644
--- a/lib/http.c
+++ b/lib/http.c
@@ -3970,18 +3970,23 @@ CURLcode Curl_bump_headersize(struct Curl_easy *data,
                               bool connect_only)
 {
   size_t bad = 0;
+  unsigned int max = MAX_HTTP_RESP_HEADER_SIZE;
   if(delta < MAX_HTTP_RESP_HEADER_SIZE) {
+    data->info.header_size += (unsigned int)delta;
+    data->req.allheadercount += (unsigned int)delta;
     if(!connect_only)
       data->req.headerbytecount += (unsigned int)delta;
-    data->info.header_size += (unsigned int)delta;
-    if(data->info.header_size > MAX_HTTP_RESP_HEADER_SIZE)
+    if(data->req.allheadercount > max)
+      bad = data->req.allheadercount;
+    else if(data->info.header_size > (max * 20)) {
       bad = data->info.header_size;
+      max *= 20;
+    }
   }
   else
-    bad = data->info.header_size + delta;
+    bad = data->req.allheadercount + delta;
   if(bad) {
-    failf(data, "Too large response headers: %zu > %u",
-          bad, MAX_HTTP_RESP_HEADER_SIZE);
+    failf(data, "Too large response headers: %zu > %u", bad, max);
     return CURLE_RECV_ERROR;
   }
   return CURLE_OK;
diff --git a/lib/urldata.h b/lib/urldata.h
index 4bfb3b48d26e5a..c6e69f3db9e90f 100644
--- a/lib/urldata.h
+++ b/lib/urldata.h
@@ -640,7 +640,9 @@ struct SingleRequest {
   curl_off_t pendingheader;      /* this many bytes left to send is actually
                                     header and not body */
   struct curltime start;         /* transfer started at this time */
-  unsigned int headerbytecount;  /* only count received headers */
+  unsigned int headerbytecount;  /* received server headers (not CONNECT
+                                    headers) */
+  unsigned int allheadercount;   /* all received headers (server + CONNECT) */
   unsigned int deductheadercount; /* this amount of bytes doesn't count when
                                      we check if anything has been transferred
                                      at the end of a connection. We use this