You could try changing the nameservers on a computer to something else (eg google's 188.8.131.52) to see if that helps. I don't think it will as your name resolution www.ea.com->184.108.40.206 already seems to be standard.
You haven't got any services such as BT Family Safety (or whatever it is called) that might be intercepting the traffic?
as bullitt asked: can you give more precise symptoms?
I can't see how it could identify you to block your connection through changed IPs. Did you check they weren't all in pretty much the same range? It might have blocked a range for some reason.
I assume you have tried different browsers? Seems a very unlikely reason, but when you've tried the sensible ones try the silly ones.
Have you tried 'ipconfig /flushdns' to clear the cache? Again, looks unlikely as your nslookup resolved it right.
Was just on twitter and Rob Hodson was stating their was problems for some people connecting to EA, perhaps just BT customers?