Where you able to download prior or has this only just occured?

As the information you have provided could be related to the network provider and the internet settings they supply, some network providers use proxys which can cause issues.

 

After updating to ICS via Sony Companion, I just encountered this same issue (however, I cannot verify the cellular network access because I don't have data) for Google Play, and in general internet access via any browser - I cannot download any of the apps I had nor update them. If I disconnect and reconnect the WiFi, it seems to work for 1 or 2 seconds, but results in the same thing thereafter. I have never encountered this problem on Gingerbread, so I don't think it's an issue with my WiFi connection, but rather ICS perhaps interfering with my WiFi connection or something along that line.

I have tried factory reset a couple of times, and even repair through PC Companion - nothing. It doesn't resolve it. I'm left with the resolve that I would need to see a Sony mobile specialist.