I've been having a similar problem, but not related to the speed of requests.
I have mine set up to pull a web page every 60 sec, and I have an LED indicate when there was a connection failure. Maybe I should log the failures so I know for sure how often they happen, but it seems to have a fair number of them throughout the day.
The server is of course up and responding, but I'm not sure what to look at in my Arduino to see what the problem is.