I recently upgraded to an iPhone 4, and was hit with this problem when trying to update apps on the phone. It has now effected every device under that account name - I can't download anything from the iTunes store.
I've emailed iTunes support, and had called Apple Support. After a few days and countless emails and phone conversations - they consolidated all the reports under the iTunes case number. This started on September 27 for me. It is now in one of the upper tier support area, where I hope it gets resolved in due time.
In my case, the security code for my credit card (same one being used for the past 5 years, and expiration date already upgraded once) was always blank and would let you enter a number, and the phone number area were always blank, but once you clicked "done" I would get the "iTunes cannot process at this time. Please try again."
From my troubleshooting - it appears to be a problem at the server end - not my devices. Use the iTunes support email and start a case if you're having this problem.