ElasticSearch / Python: CannotSendRequest error

I'm using the ElasticSearch python client to connect to an ES node. When my application is running a fairly heavy query (in the sense that it's taking a few hundred milliseconds to process), I'm getting the following error:

ConnectionError: ConnectionError(HTTPConnectionPool(host='...', port=9200):
Max retries exceeded with url: ... (Caused by 
<class 'httplib.CannotSendRequest'>: )) caused by:
MaxRetryError(HTTPConnectionPool(host='...', port=9200): 
Max retries exceeded with url: ... 
(Caused by <class 'httplib.CannotSendRequest'>: ))

My initial guess was that I'm running out of connections in the connection pool, but I've tried to increase the size of the pool and it didn't do help. So ... I'm a little lost now.

My code is really straightforward:

es = Elasticsearch(hosts=[ IP_of_a_single_node ], maxsize=25)
res = es.search("my_index", "my_doc_type", query_string)

(As I wrote above, I've tried different pool sizes = the default is 10, I've tried 25, and then even more - it made no difference).