Data Roaming turned off is simply not enough. You also have to turn off all push notifications including push email and visual voicemail. Apparently, those things bypass data roaming and charges can accrue.
That's 100% not true. I spent a week in Europe and just had data roaming off (push and everything was still on) and didn't have any roaming fees, except for a few cents when I knowingly turned on data to download an important email I needed. Turning data roaming off stops all data transfer over EDGE/3G. Period. Including visual voicemail. When you get a voicemail with roaming off, you'll get the notification but when you try to go to your voicemail, it will dial in the old fashioned way, it will not be visual. With data roaming off, when overseas the only roaming charges you will get hit with are for calls and text.
In that case, I will be calling ATT tomorrow to turn all incoming calls off. A family member, who lives in Europe, is lending me one of his cellphones, so screw you ATT -- I'm not paying your ridiculous roaming fees!
That's how every carrier works. If I'm over in the UK roaming on Vodafone or whatever and someone calls my phone, the call is first routed to the exchange my phone number belongs to, which for me is in St. Louis. The switch looks me up to find out what AT&T tower my phone is connected to so it can route it, but sees that I'm not on AT&T right now and roaming on Vodafone in the UK. It will then route the call across the pond, into Vodafone's network, which will then figure out what tower I'm on in the UK, and send the call to my phone. Whether I answer the phone or not, that's a minute of usage right there that Vodafone bills AT&T who then in turn bills the customer.
If I choose to send the call to voicemail, my phone will then forward the call to AT&T's voicemail system number. But it's not quite that simple, as the forwarded call to voicemail will route back through Vodafone's network, back across the pond in the other direction, and into AT&T's network where it will finally get to the voicemail system, and Vodafone bills AT&T for each minute the caller spends leaving a message, who then in turn bills the customer. So if someone calls you while you're roaming and leaves a voicemail, the call is routed to your location and then back to your home location. It's a horribly inefficient system, but it's just limitations of the PSTN which was designed 100 years ago when the thought of anyone having a portable phone they could use anywhere in the world was Alexander Graham Bell's wet dream. It's not the fault of one specific carrier, it's the fault of an inefficient telephone network. The carrier you're roaming on understandably wants compensation for your usage, they're not going to just give out free rides to roamers, and you're the one who gets stuck with the charges because you're the one using your phone in a foreign country.