By "breaks" I just meant I could see the intermittent GET failures. Sorry - just my slang.
No the code doesn't freeze. What happens is the GET which arrives while another GET is being processed is just not handled and there is no response.
I found this problem with another sketch which serves up fairly complex webpages with js files, images etc. It took some investigation to track down why some of the GETs from the html were never being processed.
I've looked at the Ethernet code and if this really is a bug, it will be a tricky one to fix.